LMFDB - Embedded newform 475.3.d.b.474.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.d (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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.4
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	475.474
Dual form			475.3.d.b.474.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60555 q^{2} +3.60555 q^{3} +9.00000 q^{4} +13.0000 q^{6} +5.00000i q^{7} +18.0278 q^{8} +4.00000 q^{9} +O(q^{10})$	\(q+3.60555 q^{2} +3.60555 q^{3} +9.00000 q^{4} +13.0000 q^{6} +5.00000i q^{7} +18.0278 q^{8} +4.00000 q^{9} -10.0000 q^{11} +32.4500 q^{12} -3.60555 q^{13} +18.0278i q^{14} +29.0000 q^{16} -15.0000i q^{17} +14.4222 q^{18} +(6.00000 - 18.0278i) q^{19} +18.0278i q^{21} -36.0555 q^{22} +35.0000i q^{23} +65.0000 q^{24} -13.0000 q^{26} -18.0278 q^{27} +45.0000i q^{28} -18.0278i q^{29} -36.0555i q^{31} +32.4500 q^{32} -36.0555 q^{33} -54.0833i q^{34} +36.0000 q^{36} -21.6333 q^{37} +(21.6333 - 65.0000i) q^{38} -13.0000 q^{39} +36.0555i q^{41} +65.0000i q^{42} -20.0000i q^{43} -90.0000 q^{44} +126.194i q^{46} -10.0000i q^{47} +104.561 q^{48} +24.0000 q^{49} -54.0833i q^{51} -32.4500 q^{52} +75.7166 q^{53} -65.0000 q^{54} +90.1388i q^{56} +(21.6333 - 65.0000i) q^{57} -65.0000i q^{58} -18.0278i q^{59} -40.0000 q^{61} -130.000i q^{62} +20.0000i q^{63} +1.00000 q^{64} -130.000 q^{66} +39.6611 q^{67} -135.000i q^{68} +126.194i q^{69} +108.167i q^{71} +72.1110 q^{72} +105.000i q^{73} -78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000i q^{77} -46.8722 q^{78} +36.0555i q^{79} -101.000 q^{81} +130.000i q^{82} -40.0000i q^{83} +162.250i q^{84} -72.1110i q^{86} -65.0000i q^{87} -180.278 q^{88} -18.0278i q^{91} +315.000i q^{92} -130.000i q^{93} -36.0555i q^{94} +117.000 q^{96} +122.589 q^{97} +86.5332 q^{98} -40.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9}+O(q^{10}) $ 4 * q + 36 * q^4 + 52 * q^6 + 16 * q^9	$ 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9} - 40 q^{11} + 116 q^{16} + 24 q^{19} + 260 q^{24} - 52 q^{26} + 144 q^{36} - 52 q^{39} - 360 q^{44} + 96 q^{49} - 260 q^{54} - 160 q^{61} + 4 q^{64} - 520 q^{66} - 312 q^{74} + 216 q^{76} - 404 q^{81} + 468 q^{96} - 160 q^{99}+O(q^{100}) $ 4 * q + 36 * q^4 + 52 * q^6 + 16 * q^9 - 40 * q^11 + 116 * q^16 + 24 * q^19 + 260 * q^24 - 52 * q^26 + 144 * q^36 - 52 * q^39 - 360 * q^44 + 96 * q^49 - 260 * q^54 - 160 * q^61 + 4 * q^64 - 520 * q^66 - 312 * q^74 + 216 * q^76 - 404 * q^81 + 468 * q^96 - 160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	3.60555			1.80278			0.901388	−	0.433013i	$-0.142549\pi$
$2$	3.60555			1.80278			0.901388	+	0.433013i	$0.142549\pi$
$3$	3.60555			1.20185			0.600925	−	0.799305i	$-0.294799\pi$
$3$	3.60555			1.20185			0.600925	+	0.799305i	$0.294799\pi$
$4$	9.00000			2.25000
$5$		0			0
$5$		0			0
$6$	13.0000			2.16667
$7$			5.00000i			0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$			5.00000i			0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	18.0278			2.25347
$9$	4.00000			0.444444
$10$		0			0
$11$	−10.0000			−0.909091			−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000			−0.909091			−0.454545	+	0.890724i	$0.650198\pi$
$12$	32.4500			2.70416
$13$	−3.60555			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−3.60555			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$			18.0278i			1.28770i
$15$		0			0
$16$	29.0000			1.81250
$17$		−	15.0000i		−	0.882353i	−0.897420	−	0.441176i	$-0.854561\pi$
$17$		−	15.0000i		−	0.882353i	0.897420	−	0.441176i	$-0.145439\pi$
$18$	14.4222			0.801234
$19$	6.00000	−	18.0278i	0.315789	−	0.948829i
$19$	6.00000	−	18.0278i	0.315789	−	0.948829i
$20$		0			0
$21$			18.0278i			0.858465i
$22$	−36.0555			−1.63889
$23$			35.0000i			1.52174i	0.648905	+	0.760870i	$0.275227\pi$
$23$			35.0000i			1.52174i	−0.648905	+	0.760870i	$0.724773\pi$
$24$	65.0000			2.70833
$25$		0			0
$26$	−13.0000			−0.500000
$27$	−18.0278			−0.667695
$28$			45.0000i			1.60714i
$29$		−	18.0278i		−	0.621647i	−0.950468	−	0.310823i	$-0.899395\pi$
$29$		−	18.0278i		−	0.621647i	0.950468	−	0.310823i	$-0.100605\pi$
$30$		0			0
$31$		−	36.0555i		−	1.16308i	−0.813517	−	0.581541i	$-0.802450\pi$
$31$		−	36.0555i		−	1.16308i	0.813517	−	0.581541i	$-0.197550\pi$
$32$	32.4500			1.01406
$33$	−36.0555			−1.09259
$34$		−	54.0833i		−	1.59068i
$35$		0			0
$36$	36.0000			1.00000
$37$	−21.6333			−0.584684			−0.292342	−	0.956314i	$-0.594435\pi$
$37$	−21.6333			−0.584684			−0.292342	+	0.956314i	$0.594435\pi$
$38$	21.6333	−	65.0000i	0.569298	−	1.71053i
$39$	−13.0000			−0.333333
$40$		0			0
$41$			36.0555i			0.879403i	0.898144	+	0.439701i	$0.144916\pi$
$41$			36.0555i			0.879403i	−0.898144	+	0.439701i	$0.855084\pi$
$42$			65.0000i			1.54762i
$43$		−	20.0000i		−	0.465116i	−0.972582	−	0.232558i	$-0.925290\pi$
$43$		−	20.0000i		−	0.465116i	0.972582	−	0.232558i	$-0.0747096\pi$
$44$	−90.0000			−2.04545
$45$		0			0
$46$			126.194i			2.74335i
$47$		−	10.0000i		−	0.212766i	−0.994325	−	0.106383i	$-0.966073\pi$
$47$		−	10.0000i		−	0.212766i	0.994325	−	0.106383i	$-0.0339270\pi$
$48$	104.561			2.17835
$49$	24.0000			0.489796
$50$		0			0
$51$		−	54.0833i		−	1.06046i
$52$	−32.4500			−0.624038
$53$	75.7166			1.42861			0.714307	−	0.699832i	$-0.246742\pi$
$53$	75.7166			1.42861			0.714307	+	0.699832i	$0.246742\pi$
$54$	−65.0000			−1.20370
$55$		0			0
$56$			90.1388i			1.60962i
$57$	21.6333	−	65.0000i	0.379532	−	1.14035i
$58$		−	65.0000i		−	1.12069i
$59$		−	18.0278i		−	0.305555i	−0.988261	−	0.152778i	$-0.951178\pi$
$59$		−	18.0278i		−	0.305555i	0.988261	−	0.152778i	$-0.0488218\pi$
$60$		0			0
$61$	−40.0000			−0.655738			−0.327869	−	0.944723i	$-0.606330\pi$
$61$	−40.0000			−0.655738			−0.327869	+	0.944723i	$0.606330\pi$
$62$		−	130.000i		−	2.09677i
$63$			20.0000i			0.317460i
$64$	1.00000			0.0156250
$65$		0			0
$66$	−130.000			−1.96970
$67$	39.6611			0.591956			0.295978	−	0.955195i	$-0.404354\pi$
$67$	39.6611			0.591956			0.295978	+	0.955195i	$0.404354\pi$
$68$		−	135.000i		−	1.98529i
$69$			126.194i			1.82890i
$70$		0			0
$71$			108.167i			1.52347i	0.647887	+	0.761736i	$0.275653\pi$
$71$			108.167i			1.52347i	−0.647887	+	0.761736i	$0.724347\pi$
$72$	72.1110			1.00154
$73$			105.000i			1.43836i	0.694826	+	0.719178i	$0.255481\pi$
$73$			105.000i			1.43836i	−0.694826	+	0.719178i	$0.744519\pi$
$74$	−78.0000			−1.05405
$75$		0			0
$76$	54.0000	−	162.250i	0.710526	−	2.13487i
$77$		−	50.0000i		−	0.649351i
$78$	−46.8722			−0.600925
$79$			36.0555i			0.456399i	0.973614	+	0.228199i	$0.0732838\pi$
$79$			36.0555i			0.456399i	−0.973614	+	0.228199i	$0.926716\pi$
$80$		0			0
$81$	−101.000			−1.24691
$82$			130.000i			1.58537i
$83$		−	40.0000i		−	0.481928i	−0.970534	−	0.240964i	$-0.922536\pi$
$83$		−	40.0000i		−	0.481928i	0.970534	−	0.240964i	$-0.0774635\pi$
$84$			162.250i			1.93155i
$85$		0			0
$86$		−	72.1110i		−	0.838500i
$87$		−	65.0000i		−	0.747126i
$88$	−180.278			−2.04861
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	18.0278i		−	0.198107i
$92$			315.000i			3.42391i
$93$		−	130.000i		−	1.39785i
$94$		−	36.0555i		−	0.383569i
$95$		0			0
$96$	117.000			1.21875
$97$	122.589			1.26380			0.631901	−	0.775049i	$-0.282275\pi$
$97$	122.589			1.26380			0.631901	+	0.775049i	$0.282275\pi$
$98$	86.5332			0.882992
$99$	−40.0000			−0.404040
$100$		0			0
$101$	−50.0000			−0.495050			−0.247525	−	0.968882i	$-0.579617\pi$
$101$	−50.0000			−0.495050			−0.247525	+	0.968882i	$0.579617\pi$
$102$		−	195.000i		−	1.91176i
$103$	−57.6888			−0.560086			−0.280043	−	0.959988i	$-0.590349\pi$
$103$	−57.6888			−0.560086			−0.280043	+	0.959988i	$0.590349\pi$
$104$	−65.0000			−0.625000
$105$		0			0
$106$	273.000			2.57547
$107$	75.7166			0.707632			0.353816	−	0.935315i	$-0.384884\pi$
$107$	75.7166			0.707632			0.353816	+	0.935315i	$0.384884\pi$
$108$	−162.250			−1.50231
$109$			198.305i			1.81931i	0.415359	+	0.909657i	$0.363656\pi$
$109$			198.305i			1.81931i	−0.415359	+	0.909657i	$0.636344\pi$
$110$		0			0
$111$	−78.0000			−0.702703
$112$			145.000i			1.29464i
$113$	122.589			1.08486			0.542428	−	0.840102i	$-0.317505\pi$
$113$	122.589			1.08486			0.542428	+	0.840102i	$0.317505\pi$
$114$	78.0000	−	234.361i	0.684211	−	2.05580i
$115$		0			0
$116$		−	162.250i		−	1.39871i
$117$	−14.4222			−0.123267
$118$		−	65.0000i		−	0.550847i
$119$	75.0000			0.630252
$120$		0			0
$121$	−21.0000			−0.173554
$122$	−144.222			−1.18215
$123$			130.000i			1.05691i
$124$		−	324.500i		−	2.61693i
$125$		0			0
$126$			72.1110i			0.572310i
$127$	−129.800			−1.02205			−0.511023	−	0.859567i	$-0.670733\pi$
$127$	−129.800			−1.02205			−0.511023	+	0.859567i	$0.670733\pi$
$128$	−126.194			−0.985893
$129$		−	72.1110i		−	0.559000i
$130$		0			0
$131$	112.000			0.854962			0.427481	−	0.904024i	$-0.359401\pi$
$131$	112.000			0.854962			0.427481	+	0.904024i	$0.359401\pi$
$132$	−324.500			−2.45833
$133$	90.1388	+	30.0000i	0.677735	+	0.225564i
$134$	143.000			1.06716
$135$		0			0
$136$		−	270.416i		−	1.98836i
$137$		−	125.000i		−	0.912409i	−0.889875	−	0.456204i	$-0.849209\pi$
$137$		−	125.000i		−	0.912409i	0.889875	−	0.456204i	$-0.150791\pi$
$138$			455.000i			3.29710i
$139$	−50.0000			−0.359712			−0.179856	−	0.983693i	$-0.557563\pi$
$139$	−50.0000			−0.359712			−0.179856	+	0.983693i	$0.557563\pi$
$140$		0			0
$141$		−	36.0555i		−	0.255713i
$142$			390.000i			2.74648i
$143$	36.0555			0.252136
$144$	116.000			0.805556
$145$		0			0
$146$			378.583i			2.59303i
$147$	86.5332			0.588661
$148$	−194.700			−1.31554
$149$	−70.0000			−0.469799			−0.234899	−	0.972020i	$-0.575476\pi$
$149$	−70.0000			−0.469799			−0.234899	+	0.972020i	$0.575476\pi$
$150$		0			0
$151$			36.0555i			0.238778i	0.992848	+	0.119389i	$0.0380936\pi$
$151$			36.0555i			0.238778i	−0.992848	+	0.119389i	$0.961906\pi$
$152$	108.167	−	325.000i	0.711622	−	2.13816i
$153$		−	60.0000i		−	0.392157i
$154$		−	180.278i		−	1.17063i
$155$		0			0
$156$	−117.000			−0.750000
$157$		−	10.0000i		−	0.0636943i	−0.999493	−	0.0318471i	$-0.989861\pi$
$157$		−	10.0000i		−	0.0636943i	0.999493	−	0.0318471i	$-0.0101390\pi$
$158$			130.000i			0.822785i
$159$	273.000			1.71698
$160$		0			0
$161$	−175.000			−1.08696
$162$	−364.161			−2.24791
$163$		−	270.000i		−	1.65644i	−0.560402	−	0.828221i	$-0.689353\pi$
$163$		−	270.000i		−	1.65644i	0.560402	−	0.828221i	$-0.310647\pi$
$164$			324.500i			1.97866i
$165$		0			0
$166$		−	144.222i		−	0.868808i
$167$	122.589			0.734064			0.367032	−	0.930208i	$-0.380374\pi$
$167$	122.589			0.734064			0.367032	+	0.930208i	$0.380374\pi$
$168$			325.000i			1.93452i
$169$	−156.000			−0.923077
$170$		0			0
$171$	24.0000	−	72.1110i	0.140351	−	0.421702i
$172$		−	180.000i		−	1.04651i
$173$	122.589			0.708605			0.354303	−	0.935131i	$-0.384718\pi$
$173$	122.589			0.708605			0.354303	+	0.935131i	$0.384718\pi$
$174$		−	234.361i		−	1.34690i
$175$		0			0
$176$	−290.000			−1.64773
$177$		−	65.0000i		−	0.367232i
$178$		0			0
$179$		−	36.0555i		−	0.201427i	−0.994915	−	0.100714i	$-0.967887\pi$
$179$		−	36.0555i		−	0.201427i	0.994915	−	0.100714i	$-0.0321126\pi$
$180$		0			0
$181$			108.167i			0.597605i	0.954315	+	0.298803i	$0.0965872\pi$
$181$			108.167i			0.597605i	−0.954315	+	0.298803i	$0.903413\pi$
$182$		−	65.0000i		−	0.357143i
$183$	−144.222			−0.788099
$184$			630.971i			3.42919i
$185$		0			0
$186$		−	468.722i		−	2.52001i
$187$			150.000i			0.802139i
$188$		−	90.0000i		−	0.478723i
$189$		−	90.1388i		−	0.476925i
$190$		0			0
$191$	193.000			1.01047			0.505236	−	0.862981i	$-0.331406\pi$
$191$	193.000			1.01047			0.505236	+	0.862981i	$0.331406\pi$
$192$	3.60555			0.0187789
$193$	−266.811			−1.38244			−0.691220	−	0.722645i	$-0.742926\pi$
$193$	−266.811			−1.38244			−0.691220	+	0.722645i	$0.742926\pi$
$194$	442.000			2.27835
$195$		0			0
$196$	216.000			1.10204
$197$		−	90.0000i		−	0.456853i	−0.973561	−	0.228426i	$-0.926642\pi$
$197$		−	90.0000i		−	0.456853i	0.973561	−	0.228426i	$-0.0733580\pi$
$198$	−144.222			−0.728394
$199$	−123.000			−0.618090			−0.309045	−	0.951047i	$-0.600009\pi$
$199$	−123.000			−0.618090			−0.309045	+	0.951047i	$0.600009\pi$
$200$		0			0
$201$	143.000			0.711443
$202$	−180.278			−0.892463
$203$	90.1388			0.444033
$204$		−	486.749i		−	2.38603i
$205$		0			0
$206$	−208.000			−1.00971
$207$			140.000i			0.676329i
$208$	−104.561			−0.502697
$209$	−60.0000	+	180.278i	−0.287081	+	0.862572i
$210$		0			0
$211$		−	234.361i		−	1.11071i	−0.831612	−	0.555357i	$-0.812581\pi$
$211$		−	234.361i		−	1.11071i	0.831612	−	0.555357i	$-0.187419\pi$
$212$	681.449			3.21438
$213$			390.000i			1.83099i
$214$	273.000			1.27570
$215$		0			0
$216$	−325.000			−1.50463
$217$	180.278			0.830772
$218$			715.000i			3.27982i
$219$			378.583i			1.72869i
$220$		0			0
$221$			54.0833i			0.244721i
$222$	−281.233			−1.26682
$223$	201.911			0.905430			0.452715	−	0.891655i	$-0.350456\pi$
$223$	201.911			0.905430			0.452715	+	0.891655i	$0.350456\pi$
$224$			162.250i			0.724329i
$225$		0			0
$226$	442.000			1.95575
$227$	255.994			1.12773			0.563864	−	0.825868i	$-0.309314\pi$
$227$	255.994			1.12773			0.563864	+	0.825868i	$0.309314\pi$
$228$	194.700	−	585.000i	0.853946	−	2.56579i
$229$	160.000			0.698690			0.349345	−	0.936994i	$-0.386404\pi$
$229$	160.000			0.698690			0.349345	+	0.936994i	$0.386404\pi$
$230$		0			0
$231$		−	180.278i		−	0.780422i
$232$		−	325.000i		−	1.40086i
$233$		−	270.000i		−	1.15880i	−0.815044	−	0.579399i	$-0.803287\pi$
$233$		−	270.000i		−	1.15880i	0.815044	−	0.579399i	$-0.196713\pi$
$234$	−52.0000			−0.222222
$235$		0			0
$236$		−	162.250i		−	0.687499i
$237$			130.000i			0.548523i
$238$	270.416			1.13620
$239$	−197.000			−0.824268			−0.412134	−	0.911123i	$-0.635216\pi$
$239$	−197.000			−0.824268			−0.412134	+	0.911123i	$0.635216\pi$
$240$		0			0
$241$			396.611i			1.64569i	0.568268	+	0.822844i	$0.307614\pi$
$241$			396.611i			1.64569i	−0.568268	+	0.822844i	$0.692386\pi$
$242$	−75.7166			−0.312878
$243$	−201.911			−0.830909
$244$	−360.000			−1.47541
$245$		0			0
$246$			468.722i			1.90537i
$247$	−21.6333	+	65.0000i	−0.0875842	+	0.263158i
$248$		−	650.000i		−	2.62097i
$249$		−	144.222i		−	0.579205i
$250$		0			0
$251$	−402.000			−1.60159			−0.800797	−	0.598936i	$-0.795590\pi$
$251$	−402.000			−1.60159			−0.800797	+	0.598936i	$0.795590\pi$
$252$			180.000i			0.714286i
$253$		−	350.000i		−	1.38340i
$254$	−468.000			−1.84252
$255$		0			0
$256$	−459.000			−1.79297
$257$	−418.244			−1.62741			−0.813704	−	0.581279i	$-0.802552\pi$
$257$	−418.244			−1.62741			−0.813704	+	0.581279i	$0.802552\pi$
$258$		−	260.000i		−	1.00775i
$259$		−	108.167i		−	0.417631i
$260$		0			0
$261$		−	72.1110i		−	0.276287i
$262$	403.822			1.54130
$263$			310.000i			1.17871i	0.807875	+	0.589354i	$0.200618\pi$
$263$			310.000i			1.17871i	−0.807875	+	0.589354i	$0.799382\pi$
$264$	−650.000			−2.46212
$265$		0			0
$266$	325.000	+	108.167i	1.22180	+	0.406641i
$267$		0			0
$268$	356.950			1.33190
$269$		−	108.167i		−	0.402106i	−0.979580	−	0.201053i	$-0.935564\pi$
$269$		−	108.167i		−	0.402106i	0.979580	−	0.201053i	$-0.0644364\pi$
$270$		0			0
$271$	105.000			0.387454			0.193727	−	0.981055i	$-0.437942\pi$
$271$	105.000			0.387454			0.193727	+	0.981055i	$0.437942\pi$
$272$		−	435.000i		−	1.59926i
$273$		−	65.0000i		−	0.238095i
$274$		−	450.694i		−	1.64487i
$275$		0			0
$276$			1135.75i			4.11503i
$277$			50.0000i			0.180505i	0.995919	+	0.0902527i	$0.0287675\pi$
$277$			50.0000i			0.180505i	−0.995919	+	0.0902527i	$0.971233\pi$
$278$	−180.278			−0.648480
$279$		−	144.222i		−	0.516925i
$280$		0			0
$281$		−	288.444i		−	1.02649i	−0.858242	−	0.513246i	$-0.828443\pi$
$281$		−	288.444i		−	1.02649i	0.858242	−	0.513246i	$-0.171557\pi$
$282$		−	130.000i		−	0.460993i
$283$		−	320.000i		−	1.13074i	−0.824837	−	0.565371i	$-0.808733\pi$
$283$		−	320.000i		−	1.13074i	0.824837	−	0.565371i	$-0.191267\pi$
$284$			973.499i			3.42781i
$285$		0			0
$286$	130.000			0.454545
$287$	−180.278			−0.628145
$288$	129.800			0.450694
$289$	64.0000			0.221453
$290$		0			0
$291$	442.000			1.51890
$292$			945.000i			3.23630i
$293$	219.939			0.750644			0.375322	−	0.926895i	$-0.377532\pi$
$293$	219.939			0.750644			0.375322	+	0.926895i	$0.377532\pi$
$294$	312.000			1.06122
$295$		0			0
$296$	−390.000			−1.31757
$297$	180.278			0.606995
$298$	−252.389			−0.846942
$299$		−	126.194i		−	0.422054i
$300$		0			0
$301$	100.000			0.332226
$302$			130.000i			0.430464i
$303$	−180.278			−0.594975
$304$	174.000	−	522.805i	0.572368	−	1.71975i
$305$		0			0
$306$		−	216.333i		−	0.706971i
$307$	237.966			0.775135			0.387567	−	0.921841i	$-0.373315\pi$
$307$	237.966			0.775135			0.387567	+	0.921841i	$0.373315\pi$
$308$		−	450.000i		−	1.46104i
$309$	−208.000			−0.673139
$310$		0			0
$311$	395.000			1.27010			0.635048	−	0.772472i	$-0.280980\pi$
$311$	395.000			1.27010			0.635048	+	0.772472i	$0.280980\pi$
$312$	−234.361			−0.751157
$313$			125.000i			0.399361i	0.979861	+	0.199681i	$0.0639904\pi$
$313$			125.000i			0.399361i	−0.979861	+	0.199681i	$0.936010\pi$
$314$		−	36.0555i		−	0.114826i
$315$		0			0
$316$			324.500i			1.02690i
$317$	3.60555			0.0113740			0.00568699	−	0.999984i	$-0.498190\pi$
$317$	3.60555			0.0113740			0.00568699	+	0.999984i	$0.498190\pi$
$318$	984.315			3.09533
$319$			180.278i			0.565133i
$320$		0			0
$321$	273.000			0.850467
$322$	−630.971			−1.95954
$323$	−270.416	−	90.0000i	−0.837202	−	0.278638i
$324$	−909.000			−2.80556
$325$		0			0
$326$		−	973.499i		−	2.98619i
$327$			715.000i			2.18654i
$328$			650.000i			1.98171i
$329$	50.0000			0.151976
$330$		0			0
$331$		−	198.305i		−	0.599110i	−0.954079	−	0.299555i	$-0.903162\pi$
$331$		−	198.305i		−	0.599110i	0.954079	−	0.299555i	$-0.0968382\pi$
$332$		−	360.000i		−	1.08434i
$333$	−86.5332			−0.259860
$334$	442.000			1.32335
$335$		0			0
$336$			522.805i			1.55597i
$337$	57.6888			0.171183			0.0855917	−	0.996330i	$-0.472722\pi$
$337$	57.6888			0.171183			0.0855917	+	0.996330i	$0.472722\pi$
$338$	−562.466			−1.66410
$339$	442.000			1.30383
$340$		0			0
$341$			360.555i			1.05735i
$342$	86.5332	−	260.000i	0.253021	−	0.760234i
$343$			365.000i			1.06414i
$344$		−	360.555i		−	1.04813i
$345$		0			0
$346$	442.000			1.27746
$347$			40.0000i			0.115274i	0.998338	+	0.0576369i	$0.0183566\pi$
$347$			40.0000i			0.115274i	−0.998338	+	0.0576369i	$0.981643\pi$
$348$		−	585.000i		−	1.68103i
$349$	−98.0000			−0.280802			−0.140401	−	0.990095i	$-0.544839\pi$
$349$	−98.0000			−0.280802			−0.140401	+	0.990095i	$0.544839\pi$
$350$		0			0
$351$	65.0000			0.185185
$352$	−324.500			−0.921874
$353$		−	185.000i		−	0.524079i	−0.965057	−	0.262040i	$-0.915605\pi$
$353$		−	185.000i		−	0.524079i	0.965057	−	0.262040i	$-0.0843951\pi$
$354$		−	234.361i		−	0.662036i
$355$		0			0
$356$		0			0
$357$	270.416			0.757469
$358$		−	130.000i		−	0.363128i
$359$	225.000			0.626741			0.313370	−	0.949631i	$-0.398542\pi$
$359$	225.000			0.626741			0.313370	+	0.949631i	$0.398542\pi$
$360$		0			0
$361$	−289.000	−	216.333i	−0.800554	−	0.599261i
$362$			390.000i			1.07735i
$363$	−75.7166			−0.208586
$364$		−	162.250i		−	0.445741i
$365$		0			0
$366$	−520.000			−1.42077
$367$		−	50.0000i		−	0.136240i	−0.997677	−	0.0681199i	$-0.978300\pi$
$367$		−	50.0000i		−	0.136240i	0.997677	−	0.0681199i	$-0.0217000\pi$
$368$			1015.00i			2.75815i
$369$			144.222i			0.390846i
$370$		0			0
$371$			378.583i			1.02044i
$372$		−	1170.00i		−	3.14516i
$373$	436.272			1.16963			0.584815	−	0.811167i	$-0.301167\pi$
$373$	436.272			1.16963			0.584815	+	0.811167i	$0.301167\pi$
$374$			540.833i			1.44608i
$375$		0			0
$376$		−	180.278i		−	0.479462i
$377$			65.0000i			0.172414i
$378$		−	325.000i		−	0.859788i
$379$			486.749i			1.28430i	0.766579	+	0.642150i	$0.221957\pi$
$379$			486.749i			1.28430i	−0.766579	+	0.642150i	$0.778043\pi$
$380$		0			0
$381$	−468.000			−1.22835
$382$	695.871			1.82165
$383$	−201.911			−0.527182			−0.263591	−	0.964634i	$-0.584907\pi$
$383$	−201.911			−0.527182			−0.263591	+	0.964634i	$0.584907\pi$
$384$	−455.000			−1.18490
$385$		0			0
$386$	−962.000			−2.49223
$387$		−	80.0000i		−	0.206718i
$388$	1103.30			2.84355
$389$	478.000			1.22879			0.614396	−	0.788998i	$-0.289400\pi$
$389$	478.000			1.22879			0.614396	+	0.788998i	$0.289400\pi$
$390$		0			0
$391$	525.000			1.34271
$392$	432.666			1.10374
$393$	403.822			1.02754
$394$		−	324.500i		−	0.823603i
$395$		0			0
$396$	−360.000			−0.909091
$397$		−	750.000i		−	1.88917i	−0.328269	−	0.944584i	$-0.606465\pi$
$397$		−	750.000i		−	1.88917i	0.328269	−	0.944584i	$-0.393535\pi$
$398$	−443.483			−1.11428
$399$	325.000	+	108.167i	0.814536	+	0.271094i
$400$		0			0
$401$		−	288.444i		−	0.719312i	−0.933085	−	0.359656i	$-0.882894\pi$
$401$		−	288.444i		−	0.719312i	0.933085	−	0.359656i	$-0.117106\pi$
$402$	515.594			1.28257
$403$			130.000i			0.322581i
$404$	−450.000			−1.11386
$405$		0			0
$406$	325.000			0.800493
$407$	216.333			0.531531
$408$		−	975.000i		−	2.38971i
$409$			36.0555i			0.0881553i	0.999028	+	0.0440776i	$0.0140349\pi$
$409$			36.0555i			0.0881553i	−0.999028	+	0.0440776i	$0.985965\pi$
$410$		0			0
$411$		−	450.694i		−	1.09658i
$412$	−519.199			−1.26019
$413$	90.1388			0.218254
$414$			504.777i			1.21927i
$415$		0			0
$416$	−117.000			−0.281250
$417$	−180.278			−0.432320
$418$	−216.333	+	650.000i	−0.517543	+	1.55502i
$419$	−112.000			−0.267303			−0.133652	−	0.991028i	$-0.542670\pi$
$419$	−112.000			−0.267303			−0.133652	+	0.991028i	$0.542670\pi$
$420$		0			0
$421$			630.971i			1.49874i	0.662149	+	0.749372i	$0.269645\pi$
$421$			630.971i			1.49874i	−0.662149	+	0.749372i	$0.730355\pi$
$422$		−	845.000i		−	2.00237i
$423$		−	40.0000i		−	0.0945626i
$424$	1365.00			3.21934
$425$		0			0
$426$			1406.16i			3.30086i
$427$		−	200.000i		−	0.468384i
$428$	681.449			1.59217
$429$	130.000			0.303030
$430$		0			0
$431$		−	432.666i		−	1.00387i	−0.864907	−	0.501933i	$-0.832622\pi$
$431$		−	432.666i		−	1.00387i	0.864907	−	0.501933i	$-0.167378\pi$
$432$	−522.805			−1.21020
$433$	735.532			1.69869			0.849345	−	0.527839i	$-0.176997\pi$
$433$	735.532			1.69869			0.849345	+	0.527839i	$0.176997\pi$
$434$	650.000			1.49770
$435$		0			0
$436$			1784.75i			4.09346i
$437$	630.971	+	210.000i	1.44387	+	0.480549i
$438$			1365.00i			3.11644i
$439$		−	793.221i		−	1.80688i	−0.428712	−	0.903441i	$-0.641033\pi$
$439$		−	793.221i		−	1.80688i	0.428712	−	0.903441i	$-0.358967\pi$
$440$		0			0
$441$	96.0000			0.217687
$442$			195.000i			0.441176i
$443$			670.000i			1.51242i	0.654332	+	0.756208i	$0.272950\pi$
$443$			670.000i			1.51242i	−0.654332	+	0.756208i	$0.727050\pi$
$444$	−702.000			−1.58108
$445$		0			0
$446$	728.000			1.63229
$447$	−252.389			−0.564628
$448$			5.00000i			0.0111607i
$449$			36.0555i			0.0803018i	0.999194	+	0.0401509i	$0.0127839\pi$
$449$			36.0555i			0.0803018i	−0.999194	+	0.0401509i	$0.987216\pi$
$450$		0			0
$451$		−	360.555i		−	0.799457i
$452$	1103.30			2.44093
$453$			130.000i			0.286976i
$454$	923.000			2.03304
$455$		0			0
$456$	390.000	−	1171.80i	0.855263	−	2.56975i
$457$			755.000i			1.65208i	0.563612	+	0.826039i	$0.309411\pi$
$457$			755.000i			1.65208i	−0.563612	+	0.826039i	$0.690589\pi$
$458$	576.888			1.25958
$459$			270.416i			0.589142i
$460$		0			0
$461$	772.000			1.67462			0.837310	−	0.546728i	$-0.184127\pi$
$461$	772.000			1.67462			0.837310	+	0.546728i	$0.184127\pi$
$462$		−	650.000i		−	1.40693i
$463$		−	350.000i		−	0.755940i	−0.925818	−	0.377970i	$-0.876622\pi$
$463$		−	350.000i		−	0.755940i	0.925818	−	0.377970i	$-0.123378\pi$
$464$		−	522.805i		−	1.12673i
$465$		0			0
$466$		−	973.499i		−	2.08905i
$467$		−	70.0000i		−	0.149893i	−0.997188	−	0.0749465i	$-0.976121\pi$
$467$		−	70.0000i		−	0.149893i	0.997188	−	0.0749465i	$-0.0238786\pi$
$468$	−129.800			−0.277350
$469$			198.305i			0.422826i
$470$		0			0
$471$		−	36.0555i		−	0.0765510i
$472$		−	325.000i		−	0.688559i
$473$			200.000i			0.422833i
$474$			468.722i			0.988864i
$475$		0			0
$476$	675.000			1.41807
$477$	302.866			0.634940
$478$	−710.294			−1.48597
$479$	370.000			0.772443			0.386221	−	0.922406i	$-0.373780\pi$
$479$	370.000			0.772443			0.386221	+	0.922406i	$0.373780\pi$
$480$		0			0
$481$	78.0000			0.162162
$482$			1430.00i			2.96680i
$483$	−630.971			−1.30636
$484$	−189.000			−0.390496
$485$		0			0
$486$	−728.000			−1.49794
$487$	−519.199			−1.06612			−0.533059	−	0.846078i	$-0.678958\pi$
$487$	−519.199			−1.06612			−0.533059	+	0.846078i	$0.678958\pi$
$488$	−721.110			−1.47768
$489$		−	973.499i		−	1.99080i
$490$		0			0
$491$	−632.000			−1.28717			−0.643585	−	0.765375i	$-0.722554\pi$
$491$	−632.000			−1.28717			−0.643585	+	0.765375i	$0.722554\pi$
$492$			1170.00i			2.37805i
$493$	−270.416			−0.548512
$494$	−78.0000	+	234.361i	−0.157895	+	0.474415i
$495$		0			0
$496$		−	1045.61i		−	2.10808i
$497$	−540.833			−1.08819
$498$		−	520.000i		−	1.04418i
$499$	−380.000			−0.761523			−0.380762	−	0.924673i	$-0.624338\pi$
$499$	−380.000			−0.761523			−0.380762	+	0.924673i	$0.624338\pi$
$500$		0			0
$501$	442.000			0.882236
$502$	−1449.43			−2.88731
$503$		−	45.0000i		−	0.0894632i	−0.998999	−	0.0447316i	$-0.985757\pi$
$503$		−	45.0000i		−	0.0894632i	0.998999	−	0.0447316i	$-0.0142433\pi$
$504$			360.555i			0.715387i
$505$		0			0
$506$		−	1261.94i		−	2.49396i
$507$	−562.466			−1.10940
$508$	−1168.20			−2.29960
$509$		−	829.277i		−	1.62923i	−0.580004	−	0.814614i	$-0.696949\pi$
$509$		−	829.277i		−	1.62923i	0.580004	−	0.814614i	$-0.303051\pi$
$510$		0			0
$511$	−525.000			−1.02740
$512$	−1150.17			−2.24643
$513$	−108.167	+	325.000i	−0.210851	+	0.633528i
$514$	−1508.00			−2.93385
$515$		0			0
$516$		−	648.999i		−	1.25775i
$517$			100.000i			0.193424i
$518$		−	390.000i		−	0.752896i
$519$	442.000			0.851638
$520$		0			0
$521$			612.944i			1.17648i	0.808688	+	0.588238i	$0.200178\pi$
$521$			612.944i			1.17648i	−0.808688	+	0.588238i	$0.799822\pi$
$522$		−	260.000i		−	0.498084i
$523$	−465.116			−0.889323			−0.444662	−	0.895699i	$-0.646676\pi$
$523$	−465.116			−0.889323			−0.444662	+	0.895699i	$0.646676\pi$
$524$	1008.00			1.92366
$525$		0			0
$526$			1117.72i			2.12494i
$527$	−540.833			−1.02625
$528$	−1045.61			−1.98032
$529$	−696.000			−1.31569
$530$		0			0
$531$		−	72.1110i		−	0.135802i
$532$	811.249	+	270.000i	1.52490	+	0.507519i
$533$		−	130.000i		−	0.243902i
$534$		0			0
$535$		0			0
$536$	715.000			1.33396
$537$		−	130.000i		−	0.242086i
$538$		−	390.000i		−	0.724907i
$539$	−240.000			−0.445269
$540$		0			0
$541$	−600.000			−1.10906			−0.554529	−	0.832165i	$-0.687101\pi$
$541$	−600.000			−1.10906			−0.554529	+	0.832165i	$0.687101\pi$
$542$	378.583			0.698492
$543$			390.000i			0.718232i
$544$		−	486.749i		−	0.894760i
$545$		0			0
$546$		−	234.361i		−	0.429232i
$547$	−598.522			−1.09419			−0.547095	−	0.837071i	$-0.684266\pi$
$547$	−598.522			−1.09419			−0.547095	+	0.837071i	$0.684266\pi$
$548$		−	1125.00i		−	2.05292i
$549$	−160.000			−0.291439
$550$		0			0
$551$	−325.000	−	108.167i	−0.589837	−	0.196310i
$552$			2275.00i			4.12138i
$553$	−180.278			−0.325999
$554$			180.278i			0.325411i
$555$		0			0
$556$	−450.000			−0.809353
$557$		−	380.000i		−	0.682226i	−0.940022	−	0.341113i	$-0.889196\pi$
$557$		−	380.000i		−	0.682226i	0.940022	−	0.341113i	$-0.110804\pi$
$558$		−	520.000i		−	0.931900i
$559$			72.1110i			0.129000i
$560$		0			0
$561$			540.833i			0.964051i
$562$		−	1040.00i		−	1.85053i
$563$	−122.589			−0.217742			−0.108871	−	0.994056i	$-0.534724\pi$
$563$	−122.589			−0.217742			−0.108871	+	0.994056i	$0.534724\pi$
$564$		−	324.500i		−	0.575354i
$565$		0			0
$566$		−	1153.78i		−	2.03847i
$567$		−	505.000i		−	0.890653i
$568$			1950.00i			3.43310i
$569$		−	36.0555i		−	0.0633665i	−0.999498	−	0.0316832i	$-0.989913\pi$
$569$		−	36.0555i		−	0.0633665i	0.999498	−	0.0316832i	$-0.0100868\pi$
$570$		0			0
$571$	−790.000			−1.38354			−0.691769	−	0.722119i	$-0.743168\pi$
$571$	−790.000			−1.38354			−0.691769	+	0.722119i	$0.743168\pi$
$572$	324.500			0.567307
$573$	695.871			1.21444
$574$	−650.000			−1.13240
$575$		0			0
$576$	4.00000			0.00694444
$577$		−	675.000i		−	1.16984i	−0.811090	−	0.584922i	$-0.801125\pi$
$577$		−	675.000i		−	1.16984i	0.811090	−	0.584922i	$-0.198875\pi$
$578$	230.755			0.399231
$579$	−962.000			−1.66149
$580$		0			0
$581$	200.000			0.344234
$582$	1593.65			2.73824
$583$	−757.166			−1.29874
$584$			1892.91i			3.24129i
$585$		0			0
$586$	793.000			1.35324
$587$			280.000i			0.477002i	0.971142	+	0.238501i	$0.0766560\pi$
$587$			280.000i			0.477002i	−0.971142	+	0.238501i	$0.923344\pi$
$588$	778.799			1.32449
$589$	−650.000	−	216.333i	−1.10357	−	0.367289i
$590$		0			0
$591$		−	324.500i		−	0.549069i
$592$	−627.366			−1.05974
$593$			750.000i			1.26476i	0.774660	+	0.632378i	$0.217921\pi$
$593$			750.000i			1.26476i	−0.774660	+	0.632378i	$0.782079\pi$
$594$	650.000			1.09428
$595$		0			0
$596$	−630.000			−1.05705
$597$	−443.483			−0.742852
$598$		−	455.000i		−	0.760870i
$599$		−	504.777i		−	0.842700i	−0.906898	−	0.421350i	$-0.861556\pi$
$599$		−	504.777i		−	0.842700i	0.906898	−	0.421350i	$-0.138444\pi$
$600$		0			0
$601$			612.944i			1.01987i	0.860212	+	0.509937i	$0.170331\pi$
$601$			612.944i			1.01987i	−0.860212	+	0.509937i	$0.829669\pi$
$602$	360.555			0.598929
$603$	158.644			0.263092
$604$			324.500i			0.537251i
$605$		0			0
$606$	−650.000			−1.07261
$607$	−987.921			−1.62755			−0.813774	−	0.581182i	$-0.802590\pi$
$607$	−987.921			−1.62755			−0.813774	+	0.581182i	$0.802590\pi$
$608$	194.700	−	585.000i	0.320230	−	0.962171i
$609$	325.000			0.533662
$610$		0			0
$611$			36.0555i			0.0590107i
$612$		−	540.000i		−	0.882353i
$613$		−	1200.00i		−	1.95759i	−0.204853	−	0.978793i	$-0.565672\pi$
$613$		−	1200.00i		−	1.95759i	0.204853	−	0.978793i	$-0.434328\pi$
$614$	858.000			1.39739
$615$		0			0
$616$		−	901.388i		−	1.46329i
$617$			350.000i			0.567261i	0.958934	+	0.283630i	$0.0915389\pi$
$617$			350.000i			0.567261i	−0.958934	+	0.283630i	$0.908461\pi$
$618$	−749.955			−1.21352
$619$	−560.000			−0.904685			−0.452342	−	0.891844i	$-0.649412\pi$
$619$	−560.000			−0.904685			−0.452342	+	0.891844i	$0.649412\pi$
$620$		0			0
$621$		−	630.971i		−	1.01606i
$622$	1424.19			2.28970
$623$		0			0
$624$	−377.000			−0.604167
$625$		0			0
$626$			450.694i			0.719958i
$627$	−216.333	+	650.000i	−0.345029	+	1.03668i
$628$		−	90.0000i		−	0.143312i
$629$			324.500i			0.515898i
$630$		0			0
$631$	−1050.00			−1.66403			−0.832013	−	0.554757i	$-0.812811\pi$
$631$	−1050.00			−1.66403			−0.832013	+	0.554757i	$0.812811\pi$
$632$			650.000i			1.02848i
$633$		−	845.000i		−	1.33491i
$634$	13.0000			0.0205047
$635$		0			0
$636$	2457.00			3.86321
$637$	−86.5332			−0.135845
$638$			650.000i			1.01881i
$639$			432.666i			0.677099i
$640$		0			0
$641$			1225.89i			1.91246i	0.292615	+	0.956230i	$0.405475\pi$
$641$			1225.89i			1.91246i	−0.292615	+	0.956230i	$0.594525\pi$
$642$	984.315			1.53320
$643$		−	1030.00i		−	1.60187i	−0.598754	−	0.800933i	$-0.704337\pi$
$643$		−	1030.00i		−	1.60187i	0.598754	−	0.800933i	$-0.295663\pi$
$644$	−1575.00			−2.44565
$645$		0			0
$646$	−975.000	−	324.500i	−1.50929	−	0.502321i
$647$		−	555.000i		−	0.857805i	−0.903351	−	0.428903i	$-0.858900\pi$
$647$		−	555.000i		−	0.857805i	0.903351	−	0.428903i	$-0.141100\pi$
$648$	−1820.80			−2.80988
$649$			180.278i			0.277777i
$650$		0			0
$651$	650.000			0.998464
$652$		−	2430.00i		−	3.72699i
$653$			50.0000i			0.0765697i	0.999267	+	0.0382848i	$0.0121894\pi$
$653$			50.0000i			0.0765697i	−0.999267	+	0.0382848i	$0.987811\pi$
$654$			2577.97i			3.94185i
$655$		0			0
$656$			1045.61i			1.59392i
$657$			420.000i			0.639269i
$658$	180.278			0.273978
$659$		−	198.305i		−	0.300919i	−0.988616	−	0.150459i	$-0.951925\pi$
$659$		−	198.305i		−	0.300919i	0.988616	−	0.150459i	$-0.0480752\pi$
$660$		0			0
$661$			198.305i			0.300008i	0.988685	+	0.150004i	$0.0479287\pi$
$661$			198.305i			0.300008i	−0.988685	+	0.150004i	$0.952071\pi$
$662$		−	715.000i		−	1.08006i
$663$			195.000i			0.294118i
$664$		−	721.110i		−	1.08601i
$665$		0			0
$666$	−312.000			−0.468468
$667$	630.971			0.945984
$668$	1103.30			1.65164
$669$	728.000			1.08819
$670$		0			0
$671$	400.000			0.596125
$672$			585.000i			0.870536i
$673$	−598.522			−0.889334			−0.444667	−	0.895696i	$-0.646678\pi$
$673$	−598.522			−0.889334			−0.444667	+	0.895696i	$0.646678\pi$
$674$	208.000			0.308605
$675$		0			0
$676$	−1404.00			−2.07692
$677$	68.5055			0.101190			0.0505949	−	0.998719i	$-0.483888\pi$
$677$	68.5055			0.101190			0.0505949	+	0.998719i	$0.483888\pi$
$678$	1593.65			2.35052
$679$			612.944i			0.902715i
$680$		0			0
$681$	923.000			1.35536
$682$			1300.00i			1.90616i
$683$	237.966			0.348413			0.174207	−	0.984709i	$-0.444264\pi$
$683$	237.966			0.348413			0.174207	+	0.984709i	$0.444264\pi$
$684$	216.000	−	648.999i	0.315789	−	0.948829i
$685$		0			0
$686$			1316.03i			1.91841i
$687$	576.888			0.839721
$688$		−	580.000i		−	0.843023i
$689$	−273.000			−0.396226
$690$		0			0
$691$	−820.000			−1.18669			−0.593343	−	0.804950i	$-0.702192\pi$
$691$	−820.000			−1.18669			−0.593343	+	0.804950i	$0.702192\pi$
$692$	1103.30			1.59436
$693$		−	200.000i		−	0.288600i
$694$			144.222i			0.207813i
$695$		0			0
$696$		−	1171.80i		−	1.68363i
$697$	540.833			0.775944
$698$	−353.344			−0.506224
$699$		−	973.499i		−	1.39270i
$700$		0			0
$701$	−540.000			−0.770328			−0.385164	−	0.922848i	$-0.625855\pi$
$701$	−540.000			−0.770328			−0.385164	+	0.922848i	$0.625855\pi$
$702$	234.361			0.333847
$703$	−129.800	+	390.000i	−0.184637	+	0.554765i
$704$	−10.0000			−0.0142045
$705$		0			0
$706$		−	667.027i		−	0.944797i
$707$		−	250.000i		−	0.353607i
$708$		−	585.000i		−	0.826271i
$709$	−268.000			−0.377997			−0.188999	−	0.981977i	$-0.560524\pi$
$709$	−268.000			−0.377997			−0.188999	+	0.981977i	$0.560524\pi$
$710$		0			0
$711$			144.222i			0.202844i
$712$		0			0
$713$	1261.94			1.76991
$714$	975.000			1.36555
$715$		0			0
$716$		−	324.500i		−	0.453212i
$717$	−710.294			−0.990647
$718$	811.249			1.12987
$719$	−105.000			−0.146036			−0.0730181	−	0.997331i	$-0.523263\pi$
$719$	−105.000			−0.146036			−0.0730181	+	0.997331i	$0.523263\pi$
$720$		0			0
$721$		−	288.444i		−	0.400061i
$722$	−1042.00	−	780.000i	−1.44322	−	1.08033i
$723$			1430.00i			1.97787i
$724$			973.499i			1.34461i
$725$		0			0
$726$	−273.000			−0.376033
$727$		−	695.000i		−	0.955983i	−0.878364	−	0.477992i	$-0.841365\pi$
$727$		−	695.000i		−	0.955983i	0.878364	−	0.477992i	$-0.158635\pi$
$728$		−	325.000i		−	0.446429i
$729$	181.000			0.248285
$730$		0			0
$731$	−300.000			−0.410397
$732$	−1298.00			−1.77322
$733$			160.000i			0.218281i	0.994026	+	0.109141i	$0.0348098\pi$
$733$			160.000i			0.218281i	−0.994026	+	0.109141i	$0.965190\pi$
$734$		−	180.278i		−	0.245610i
$735$		0			0
$736$			1135.75i			1.54314i
$737$	−396.611			−0.538142
$738$			520.000i			0.704607i
$739$	−1028.00			−1.39107			−0.695535	−	0.718493i	$-0.744832\pi$
$739$	−1028.00			−1.39107			−0.695535	+	0.718493i	$0.744832\pi$
$740$		0			0
$741$	−78.0000	+	234.361i	−0.105263	+	0.316276i
$742$			1365.00i			1.83962i
$743$	526.410			0.708493			0.354247	−	0.935152i	$-0.384737\pi$
$743$	526.410			0.708493			0.354247	+	0.935152i	$0.384737\pi$
$744$		−	2343.61i		−	3.15001i
$745$		0			0
$746$	1573.00			2.10858
$747$		−	160.000i		−	0.214190i
$748$			1350.00i			1.80481i
$749$			378.583i			0.505451i
$750$		0			0
$751$			36.0555i			0.0480100i	0.999712	+	0.0240050i	$0.00764176\pi$
$751$			36.0555i			0.0480100i	−0.999712	+	0.0240050i	$0.992358\pi$
$752$		−	290.000i		−	0.385638i
$753$	−1449.43			−1.92488
$754$			234.361i			0.310823i
$755$		0			0
$756$		−	811.249i		−	1.07308i
$757$		−	60.0000i		−	0.0792602i	−0.999214	−	0.0396301i	$-0.987382\pi$
$757$		−	60.0000i		−	0.0792602i	0.999214	−	0.0396301i	$-0.0126180\pi$
$758$			1755.00i			2.31530i
$759$		−	1261.94i		−	1.66264i
$760$		0			0
$761$	−655.000			−0.860710			−0.430355	−	0.902660i	$-0.641612\pi$
$761$	−655.000			−0.860710			−0.430355	+	0.902660i	$0.641612\pi$
$762$	−1687.40			−2.21443
$763$	−991.527			−1.29951
$764$	1737.00			2.27356
$765$		0			0
$766$	−728.000			−0.950392
$767$			65.0000i			0.0847458i
$768$	−1654.95			−2.15488
$769$	185.000			0.240572			0.120286	−	0.992739i	$-0.461619\pi$
$769$	185.000			0.240572			0.120286	+	0.992739i	$0.461619\pi$
$770$		0			0
$771$	−1508.00			−1.95590
$772$	−2401.30			−3.11049
$773$	320.894			0.415128			0.207564	−	0.978221i	$-0.433446\pi$
$773$	320.894			0.415128			0.207564	+	0.978221i	$0.433446\pi$
$774$		−	288.444i		−	0.372667i
$775$		0			0
$776$	2210.00			2.84794
$777$		−	390.000i		−	0.501931i
$778$	1723.45			2.21524
$779$	650.000	+	216.333i	0.834403	+	0.277706i
$780$		0			0
$781$		−	1081.67i		−	1.38497i
$782$	1892.91			2.42061
$783$			325.000i			0.415070i
$784$	696.000			0.887755
$785$		0			0
$786$	1456.00			1.85242
$787$	68.5055			0.0870463			0.0435232	−	0.999052i	$-0.486142\pi$
$787$	68.5055			0.0870463			0.0435232	+	0.999052i	$0.486142\pi$
$788$		−	810.000i		−	1.02792i
$789$			1117.72i			1.41663i
$790$		0			0
$791$			612.944i			0.774897i
$792$	−721.110			−0.910493
$793$	144.222			0.181869
$794$		−	2704.16i		−	3.40575i
$795$		0			0
$796$	−1107.00			−1.39070
$797$	1258.34			1.57884			0.789421	−	0.613852i	$-0.210381\pi$
$797$	1258.34			1.57884			0.789421	+	0.613852i	$0.210381\pi$
$798$	1171.80	+	390.000i	1.46843	+	0.488722i
$799$	−150.000			−0.187735
$800$		0			0
$801$		0			0
$802$		−	1040.00i		−	1.29676i
$803$		−	1050.00i		−	1.30760i
$804$	1287.00			1.60075
$805$		0			0
$806$			468.722i			0.581541i
$807$		−	390.000i		−	0.483271i
$808$	−901.388			−1.11558
$809$	7.00000			0.00865266			0.00432633	−	0.999991i	$-0.498623\pi$
$809$	7.00000			0.00865266			0.00432633	+	0.999991i	$0.498623\pi$
$810$		0			0
$811$			1027.58i			1.26706i	0.773720	+	0.633528i	$0.218394\pi$
$811$			1027.58i			1.26706i	−0.773720	+	0.633528i	$0.781606\pi$
$812$	811.249			0.999075
$813$	378.583			0.465662
$814$	780.000			0.958231
$815$		0			0
$816$		−	1568.41i		−	1.92208i
$817$	−360.555	−	120.000i	−0.441316	−	0.146879i
$818$			130.000i			0.158924i
$819$		−	72.1110i		−	0.0880477i
$820$		0			0
$821$	842.000			1.02558			0.512789	−	0.858515i	$-0.328612\pi$
$821$	842.000			1.02558			0.512789	+	0.858515i	$0.328612\pi$
$822$		−	1625.00i		−	1.97689i
$823$			775.000i			0.941677i	0.882219	+	0.470838i	$0.156049\pi$
$823$			775.000i			0.941677i	−0.882219	+	0.470838i	$0.843951\pi$
$824$	−1040.00			−1.26214
$825$		0			0
$826$	325.000			0.393462
$827$	−1265.55			−1.53029			−0.765144	−	0.643859i	$-0.777332\pi$
$827$	−1265.55			−1.53029			−0.765144	+	0.643859i	$0.777332\pi$
$828$			1260.00i			1.52174i
$829$			1568.41i			1.89194i	0.324260	+	0.945968i	$0.394885\pi$
$829$			1568.41i			1.89194i	−0.324260	+	0.945968i	$0.605115\pi$
$830$		0			0
$831$			180.278i			0.216941i
$832$	−3.60555			−0.00433360
$833$		−	360.000i		−	0.432173i
$834$	−650.000			−0.779376
$835$		0			0
$836$	−540.000	+	1622.50i	−0.645933	+	1.94079i
$837$			650.000i			0.776583i
$838$	−403.822			−0.481888
$839$			1153.78i			1.37518i	0.726099	+	0.687590i	$0.241331\pi$
$839$			1153.78i			1.37518i	−0.726099	+	0.687590i	$0.758669\pi$
$840$		0			0
$841$	516.000			0.613555
$842$			2275.00i			2.70190i
$843$		−	1040.00i		−	1.23369i
$844$		−	2109.25i		−	2.49911i
$845$		0			0
$846$		−	144.222i		−	0.170475i
$847$		−	105.000i		−	0.123967i
$848$	2195.78			2.58936
$849$		−	1153.78i		−	1.35898i
$850$		0			0
$851$		−	757.166i		−	0.889737i
$852$			3510.00i			4.11972i
$853$			400.000i			0.468933i	0.972124	+	0.234467i	$0.0753343\pi$
$853$			400.000i			0.468933i	−0.972124	+	0.234467i	$0.924666\pi$
$854$		−	721.110i		−	0.844391i
$855$		0			0
$856$	1365.00			1.59463
$857$	158.644			0.185116			0.0925579	−	0.995707i	$-0.470496\pi$
$857$	158.644			0.185116			0.0925579	+	0.995707i	$0.470496\pi$
$858$	468.722			0.546296
$859$	−1432.00			−1.66705			−0.833527	−	0.552478i	$-0.813682\pi$
$859$	−1432.00			−1.66705			−0.833527	+	0.552478i	$0.813682\pi$
$860$		0			0
$861$	−650.000			−0.754936
$862$		−	1560.00i		−	1.80974i
$863$	129.800			0.150405			0.0752027	−	0.997168i	$-0.476040\pi$
$863$	129.800			0.150405			0.0752027	+	0.997168i	$0.476040\pi$
$864$	−585.000			−0.677083
$865$		0			0
$866$	2652.00			3.06236
$867$	230.755			0.266154
$868$	1622.50			1.86924
$869$		−	360.555i		−	0.414908i
$870$		0			0
$871$	−143.000			−0.164179
$872$			3575.00i			4.09977i
$873$	490.355			0.561690
$874$	2275.00	+	757.166i	2.60297	+	0.866322i
$875$		0			0
$876$			3407.25i			3.88955i
$877$	393.005			0.448124			0.224062	−	0.974575i	$-0.428068\pi$
$877$	393.005			0.448124			0.224062	+	0.974575i	$0.428068\pi$
$878$		−	2860.00i		−	3.25740i
$879$	793.000			0.902162
$880$		0			0
$881$	750.000			0.851305			0.425653	−	0.904887i	$-0.360045\pi$
$881$	750.000			0.851305			0.425653	+	0.904887i	$0.360045\pi$
$882$	346.133			0.392441
$883$			120.000i			0.135900i	0.997689	+	0.0679502i	$0.0216459\pi$
$883$			120.000i			0.135900i	−0.997689	+	0.0679502i	$0.978354\pi$
$884$			486.749i			0.550622i
$885$		0			0
$886$			2415.72i			2.72655i
$887$	850.910			0.959312			0.479656	−	0.877457i	$-0.340761\pi$
$887$	850.910			0.959312			0.479656	+	0.877457i	$0.340761\pi$
$888$	−1406.16			−1.58352
$889$		−	648.999i		−	0.730033i
$890$		0			0
$891$	1010.00			1.13356
$892$	1817.20			2.03722
$893$	−180.278	−	60.0000i	−0.201879	−	0.0671892i
$894$	−910.000			−1.01790
$895$		0			0
$896$		−	630.971i		−	0.704209i
$897$		−	455.000i		−	0.507246i
$898$			130.000i			0.144766i
$899$	−650.000			−0.723026
$900$		0			0
$901$		−	1135.75i		−	1.26054i
$902$		−	1300.00i		−	1.44124i
$903$	360.555			0.399286
$904$	2210.00			2.44469
$905$		0			0
$906$			468.722i			0.517353i
$907$	−645.394			−0.711570			−0.355785	−	0.934568i	$-0.615786\pi$
$907$	−645.394			−0.711570			−0.355785	+	0.934568i	$0.615786\pi$
$908$	2303.95			2.53739
$909$	−200.000			−0.220022
$910$		0			0
$911$		−	1225.89i		−	1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$911$		−	1225.89i		−	1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$912$	627.366	−	1885.00i	0.687901	−	2.06689i
$913$			400.000i			0.438116i
$914$			2722.19i			2.97833i
$915$		0			0
$916$	1440.00			1.57205
$917$			560.000i			0.610687i
$918$			975.000i			1.06209i
$919$	513.000			0.558215			0.279108	−	0.960260i	$-0.409961\pi$
$919$	513.000			0.558215			0.279108	+	0.960260i	$0.409961\pi$
$920$		0			0
$921$	858.000			0.931596
$922$	2783.49			3.01896
$923$		−	390.000i		−	0.422535i
$924$		−	1622.50i		−	1.75595i
$925$		0			0
$926$		−	1261.94i		−	1.36279i
$927$	−230.755			−0.248927
$928$		−	585.000i		−	0.630388i
$929$	1217.00			1.31001			0.655005	−	0.755624i	$-0.272666\pi$
$929$	1217.00			1.31001			0.655005	+	0.755624i	$0.272666\pi$
$930$		0			0
$931$	144.000	−	432.666i	0.154672	−	0.464733i
$932$		−	2430.00i		−	2.60730i
$933$	1424.19			1.52647
$934$		−	252.389i		−	0.270223i
$935$		0			0
$936$	−260.000			−0.277778
$937$			1095.00i			1.16862i	0.811529	+	0.584312i	$0.198635\pi$
$937$			1095.00i			1.16862i	−0.811529	+	0.584312i	$0.801365\pi$
$938$			715.000i			0.762260i
$939$			450.694i			0.479972i
$940$		0			0
$941$			1099.69i			1.16864i	0.811522	+	0.584322i	$0.198639\pi$
$941$			1099.69i			1.16864i	−0.811522	+	0.584322i	$0.801361\pi$
$942$		−	130.000i		−	0.138004i
$943$	−1261.94			−1.33822
$944$		−	522.805i		−	0.553819i
$945$		0			0
$946$			721.110i			0.762273i
$947$		−	640.000i		−	0.675818i	−0.941179	−	0.337909i	$-0.890280\pi$
$947$		−	640.000i		−	0.675818i	0.941179	−	0.337909i	$-0.109720\pi$
$948$			1170.00i			1.23418i
$949$		−	378.583i		−	0.398928i
$950$		0			0
$951$	13.0000			0.0136698
$952$	1352.08			1.42025
$953$	−1168.20			−1.22581			−0.612906	−	0.790156i	$-0.710000\pi$
$953$	−1168.20			−1.22581			−0.612906	+	0.790156i	$0.710000\pi$
$954$	1092.00			1.14465
$955$		0			0
$956$	−1773.00			−1.85460
$957$			650.000i			0.679206i
$958$	1334.05			1.39254
$959$	625.000			0.651721
$960$		0			0
$961$	−339.000			−0.352758
$962$	281.233			0.292342
$963$	302.866			0.314503
$964$			3569.50i			3.70280i
$965$		0			0
$966$	−2275.00			−2.35507
$967$			1010.00i			1.04447i	0.852802	+	0.522234i	$0.174901\pi$
$967$			1010.00i			1.04447i	−0.852802	+	0.522234i	$0.825099\pi$
$968$	−378.583			−0.391098
$969$	−975.000	−	324.500i	−1.00619	−	0.334881i
$970$		0			0
$971$		−	757.166i		−	0.779779i	−0.920862	−	0.389890i	$-0.872513\pi$
$971$		−	757.166i		−	0.779779i	0.920862	−	0.389890i	$-0.127487\pi$
$972$	−1817.20			−1.86955
$973$		−	250.000i		−	0.256937i
$974$	−1872.00			−1.92197
$975$		0			0
$976$	−1160.00			−1.18852
$977$	598.522			0.612612			0.306306	−	0.951933i	$-0.400907\pi$
$977$	598.522			0.612612			0.306306	+	0.951933i	$0.400907\pi$
$978$		−	3510.00i		−	3.58896i
$979$		0			0
$980$		0			0
$981$			793.221i			0.808584i
$982$	−2278.71			−2.32048
$983$	−57.6888			−0.0586865			−0.0293432	−	0.999569i	$-0.509342\pi$
$983$	−57.6888			−0.0586865			−0.0293432	+	0.999569i	$0.509342\pi$
$984$			2343.61i			2.38172i
$985$		0			0
$986$	−975.000			−0.988844
$987$	180.278			0.182652
$988$	−194.700	+	585.000i	−0.197065	+	0.592105i
$989$	700.000			0.707786
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		−	1170.00i		−	1.17944i
$993$		−	715.000i		−	0.720040i
$994$	−1950.00			−1.96177
$995$		0			0
$996$		−	1298.00i		−	1.30321i
$997$			170.000i			0.170512i	0.996359	+	0.0852558i	$0.0271707\pi$
$997$			170.000i			0.170512i	−0.996359	+	0.0852558i	$0.972829\pi$
$998$	−1370.11			−1.37286
$999$	390.000			0.390390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.b.474.4		4
5.2	odd	4		19.3.b.b.18.2	yes	2
5.3	odd	4		475.3.c.b.151.1		2
5.4	even	2	inner	475.3.d.b.474.1		4
15.2	even	4		171.3.c.b.37.1		2
19.18	odd	2	inner	475.3.d.b.474.2		4
20.7	even	4		304.3.e.d.113.2		2
40.27	even	4		1216.3.e.h.1025.1		2
40.37	odd	4		1216.3.e.g.1025.2		2
60.47	odd	4		2736.3.o.d.721.2		2
95.2	even	36		361.3.f.d.262.1		12
95.7	odd	12		361.3.d.b.293.1		4
95.12	even	12		361.3.d.b.293.2		4
95.17	odd	36		361.3.f.d.262.2		12
95.18	even	4		475.3.c.b.151.2		2
95.22	even	36		361.3.f.d.333.2		12
95.27	even	12		361.3.d.b.69.1		4
95.32	even	36		361.3.f.d.116.1		12
95.37	even	4		19.3.b.b.18.1	✓	2
95.42	odd	36		361.3.f.d.307.1		12
95.47	odd	36		361.3.f.d.299.1		12
95.52	even	36		361.3.f.d.127.1		12
95.62	odd	36		361.3.f.d.127.2		12
95.67	even	36		361.3.f.d.299.2		12
95.72	even	36		361.3.f.d.307.2		12
95.82	odd	36		361.3.f.d.116.2		12
95.87	odd	12		361.3.d.b.69.2		4
95.92	odd	36		361.3.f.d.333.1		12
95.94	odd	2	inner	475.3.d.b.474.3		4
285.227	odd	4		171.3.c.b.37.2		2
380.227	odd	4		304.3.e.d.113.1		2
760.37	even	4		1216.3.e.g.1025.1		2
760.227	odd	4		1216.3.e.h.1025.2		2
1140.227	even	4		2736.3.o.d.721.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.b.18.1	✓	2	95.37	even	4
19.3.b.b.18.2	yes	2	5.2	odd	4
171.3.c.b.37.1		2	15.2	even	4
171.3.c.b.37.2		2	285.227	odd	4
304.3.e.d.113.1		2	380.227	odd	4
304.3.e.d.113.2		2	20.7	even	4
361.3.d.b.69.1		4	95.27	even	12
361.3.d.b.69.2		4	95.87	odd	12
361.3.d.b.293.1		4	95.7	odd	12
361.3.d.b.293.2		4	95.12	even	12
361.3.f.d.116.1		12	95.32	even	36
361.3.f.d.116.2		12	95.82	odd	36
361.3.f.d.127.1		12	95.52	even	36
361.3.f.d.127.2		12	95.62	odd	36
361.3.f.d.262.1		12	95.2	even	36
361.3.f.d.262.2		12	95.17	odd	36
361.3.f.d.299.1		12	95.47	odd	36
361.3.f.d.299.2		12	95.67	even	36
361.3.f.d.307.1		12	95.42	odd	36
361.3.f.d.307.2		12	95.72	even	36
361.3.f.d.333.1		12	95.92	odd	36
361.3.f.d.333.2		12	95.22	even	36
475.3.c.b.151.1		2	5.3	odd	4
475.3.c.b.151.2		2	95.18	even	4
475.3.d.b.474.1		4	5.4	even	2	inner
475.3.d.b.474.2		4	19.18	odd	2	inner
475.3.d.b.474.3		4	95.94	odd	2	inner
475.3.d.b.474.4		4	1.1	even	1	trivial
1216.3.e.g.1025.1		2	760.37	even	4
1216.3.e.g.1025.2		2	40.37	odd	4
1216.3.e.h.1025.1		2	40.27	even	4
1216.3.e.h.1025.2		2	760.227	odd	4
2736.3.o.d.721.1		2	1140.227	even	4
2736.3.o.d.721.2		2	60.47	odd	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.60555 q^{2} +3.60555 q^{3} +9.00000 q^{4} +13.0000 q^{6} +5.00000i q^{7} +18.0278 q^{8} +4.00000 q^{9} +O(q^{10})\)	\(q+3.60555 q^{2} +3.60555 q^{3} +9.00000 q^{4} +13.0000 q^{6} +5.00000i q^{7} +18.0278 q^{8} +4.00000 q^{9} -10.0000 q^{11} +32.4500 q^{12} -3.60555 q^{13} +18.0278i q^{14} +29.0000 q^{16} -15.0000i q^{17} +14.4222 q^{18} +(6.00000 - 18.0278i) q^{19} +18.0278i q^{21} -36.0555 q^{22} +35.0000i q^{23} +65.0000 q^{24} -13.0000 q^{26} -18.0278 q^{27} +45.0000i q^{28} -18.0278i q^{29} -36.0555i q^{31} +32.4500 q^{32} -36.0555 q^{33} -54.0833i q^{34} +36.0000 q^{36} -21.6333 q^{37} +(21.6333 - 65.0000i) q^{38} -13.0000 q^{39} +36.0555i q^{41} +65.0000i q^{42} -20.0000i q^{43} -90.0000 q^{44} +126.194i q^{46} -10.0000i q^{47} +104.561 q^{48} +24.0000 q^{49} -54.0833i q^{51} -32.4500 q^{52} +75.7166 q^{53} -65.0000 q^{54} +90.1388i q^{56} +(21.6333 - 65.0000i) q^{57} -65.0000i q^{58} -18.0278i q^{59} -40.0000 q^{61} -130.000i q^{62} +20.0000i q^{63} +1.00000 q^{64} -130.000 q^{66} +39.6611 q^{67} -135.000i q^{68} +126.194i q^{69} +108.167i q^{71} +72.1110 q^{72} +105.000i q^{73} -78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000i q^{77} -46.8722 q^{78} +36.0555i q^{79} -101.000 q^{81} +130.000i q^{82} -40.0000i q^{83} +162.250i q^{84} -72.1110i q^{86} -65.0000i q^{87} -180.278 q^{88} -18.0278i q^{91} +315.000i q^{92} -130.000i q^{93} -36.0555i q^{94} +117.000 q^{96} +122.589 q^{97} +86.5332 q^{98} -40.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9}+O(q^{10}) \) 4 * q + 36 * q^4 + 52 * q^6 + 16 * q^9	\( 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9} - 40 q^{11} + 116 q^{16} + 24 q^{19} + 260 q^{24} - 52 q^{26} + 144 q^{36} - 52 q^{39} - 360 q^{44} + 96 q^{49} - 260 q^{54} - 160 q^{61} + 4 q^{64} - 520 q^{66} - 312 q^{74} + 216 q^{76} - 404 q^{81} + 468 q^{96} - 160 q^{99}+O(q^{100}) \) 4 * q + 36 * q^4 + 52 * q^6 + 16 * q^9 - 40 * q^11 + 116 * q^16 + 24 * q^19 + 260 * q^24 - 52 * q^26 + 144 * q^36 - 52 * q^39 - 360 * q^44 + 96 * q^49 - 260 * q^54 - 160 * q^61 + 4 * q^64 - 520 * q^66 - 312 * q^74 + 216 * q^76 - 404 * q^81 + 468 * q^96 - 160 * q^99

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