LMFDB - Embedded newform 405.4.b.b.244.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(244,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{6} + 567x^{4} + 2024x^{2} + 1900 $ x^8 + 44x^6 + 567x^4 + 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.7
Root			$3.99806i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.b.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.99806i q^{2} -7.98447 q^{4} +(3.61252 + 10.5806i) q^{5} -7.79840i q^{7} +0.0620886i q^{8} +O(q^{10})$	\(q+3.99806i q^{2} -7.98447 q^{4} +(3.61252 + 10.5806i) q^{5} -7.79840i q^{7} +0.0620886i q^{8} +(-42.3020 + 14.4431i) q^{10} -32.7050 q^{11} -65.9775i q^{13} +31.1784 q^{14} -64.1240 q^{16} +15.5853i q^{17} -51.6661 q^{19} +(-28.8440 - 84.4807i) q^{20} -130.756i q^{22} +49.6796i q^{23} +(-98.8994 + 76.4454i) q^{25} +263.782 q^{26} +62.2661i q^{28} -282.020 q^{29} -147.663 q^{31} -255.875i q^{32} -62.3111 q^{34} +(82.5119 - 28.1718i) q^{35} -122.498i q^{37} -206.564i q^{38} +(-0.656937 + 0.224296i) q^{40} +109.032 q^{41} +191.493i q^{43} +261.132 q^{44} -198.622 q^{46} -414.420i q^{47} +282.185 q^{49} +(-305.633 - 395.406i) q^{50} +526.796i q^{52} -236.381i q^{53} +(-118.147 - 346.039i) q^{55} +0.484192 q^{56} -1127.53i q^{58} +784.024 q^{59} -535.138 q^{61} -590.363i q^{62} +510.010 q^{64} +(698.084 - 238.345i) q^{65} -121.069i q^{67} -124.441i q^{68} +(112.633 + 329.888i) q^{70} -590.432 q^{71} +986.078i q^{73} +489.753 q^{74} +412.526 q^{76} +255.046i q^{77} +607.025 q^{79} +(-231.649 - 678.472i) q^{80} +435.915i q^{82} +1292.46i q^{83} +(-164.903 + 56.3023i) q^{85} -765.598 q^{86} -2.03061i q^{88} -1506.78 q^{89} -514.519 q^{91} -396.665i q^{92} +1656.87 q^{94} +(-186.645 - 546.659i) q^{95} +610.887i q^{97} +1128.19i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} - 15 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 - 15 * q^5	$ 8 q - 24 q^{4} - 15 q^{5} + 7 q^{10} + 42 q^{14} + 4 q^{16} + 118 q^{19} + 129 q^{20} + 17 q^{25} + 36 q^{26} - 318 q^{29} - 416 q^{31} + 638 q^{34} - 192 q^{35} - 265 q^{40} - 486 q^{41} - 852 q^{44} - 598 q^{46} + 350 q^{49} - 1143 q^{50} - 162 q^{55} - 1530 q^{56} + 1146 q^{59} - 398 q^{61} + 1640 q^{64} + 1833 q^{65} + 630 q^{70} + 1728 q^{71} - 1218 q^{74} - 3498 q^{76} + 2596 q^{79} - 1923 q^{80} - 233 q^{85} - 480 q^{86} - 1086 q^{89} - 2574 q^{91} + 1238 q^{94} - 1674 q^{95}+O(q^{100}) $ 8 * q - 24 * q^4 - 15 * q^5 + 7 * q^10 + 42 * q^14 + 4 * q^16 + 118 * q^19 + 129 * q^20 + 17 * q^25 + 36 * q^26 - 318 * q^29 - 416 * q^31 + 638 * q^34 - 192 * q^35 - 265 * q^40 - 486 * q^41 - 852 * q^44 - 598 * q^46 + 350 * q^49 - 1143 * q^50 - 162 * q^55 - 1530 * q^56 + 1146 * q^59 - 398 * q^61 + 1640 * q^64 + 1833 * q^65 + 630 * q^70 + 1728 * q^71 - 1218 * q^74 - 3498 * q^76 + 2596 * q^79 - 1923 * q^80 - 233 * q^85 - 480 * q^86 - 1086 * q^89 - 2574 * q^91 + 1238 * q^94 - 1674 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$			3.99806i			1.41353i	0.707450	+	0.706764i	$0.249846\pi$
$2$			3.99806i			1.41353i	−0.707450	+	0.706764i	$0.750154\pi$
$3$		0			0
$3$		0			0
$4$	−7.98447			−0.998059
$5$	3.61252	+	10.5806i	0.323113	+	0.946360i
$5$	3.61252	+	10.5806i	0.323113	+	0.946360i
$6$		0			0
$7$		−	7.79840i		−	0.421074i	−0.977586	−	0.210537i	$-0.932479\pi$
$7$		−	7.79840i		−	0.421074i	0.977586	−	0.210537i	$-0.0675212\pi$
$8$			0.0620886i			0.00274396i
$9$		0			0
$10$	−42.3020	+	14.4431i	−1.33771	+	0.456730i
$11$	−32.7050			−0.896448			−0.448224	−	0.893921i	$-0.647943\pi$
$11$	−32.7050			−0.896448			−0.448224	+	0.893921i	$0.647943\pi$
$12$		0			0
$13$		−	65.9775i		−	1.40761i	−0.710395	−	0.703803i	$-0.751484\pi$
$13$		−	65.9775i		−	1.40761i	0.710395	−	0.703803i	$-0.248516\pi$
$14$	31.1784			0.595199
$15$		0			0
$16$	−64.1240			−1.00194
$17$			15.5853i			0.222353i	0.993801	+	0.111176i	$0.0354619\pi$
$17$			15.5853i			0.222353i	−0.993801	+	0.111176i	$0.964538\pi$
$18$		0			0
$19$	−51.6661			−0.623843			−0.311921	−	0.950108i	$-0.600973\pi$
$19$	−51.6661			−0.623843			−0.311921	+	0.950108i	$0.600973\pi$
$20$	−28.8440	−	84.4807i	−0.322486	−	0.944523i
$21$		0			0
$22$		−	130.756i		−	1.26715i
$23$			49.6796i			0.450387i	0.974314	+	0.225194i	$0.0723015\pi$
$23$			49.6796i			0.450387i	−0.974314	+	0.225194i	$0.927699\pi$
$24$		0			0
$25$	−98.8994	+	76.4454i	−0.791195	+	0.611563i
$26$	263.782			1.98969
$27$		0			0
$28$			62.2661i			0.420256i
$29$	−282.020			−1.80586			−0.902929	−	0.429790i	$-0.858587\pi$
$29$	−282.020			−1.80586			−0.902929	+	0.429790i	$0.858587\pi$
$30$		0			0
$31$	−147.663			−0.855515			−0.427758	−	0.903893i	$-0.640696\pi$
$31$	−147.663			−0.855515			−0.427758	+	0.903893i	$0.640696\pi$
$32$		−	255.875i		−	1.41352i
$33$		0			0
$34$	−62.3111			−0.314302
$35$	82.5119	−	28.1718i	0.398488	−	0.136055i
$36$		0			0
$37$		−	122.498i		−	0.544284i	−0.962257	−	0.272142i	$-0.912268\pi$
$37$		−	122.498i		−	0.544284i	0.962257	−	0.272142i	$-0.0877320\pi$
$38$		−	206.564i		−	0.881818i
$39$		0			0
$40$	−0.656937	+	0.224296i	−0.00259677	+	0.000886609i
$41$	109.032			0.415314			0.207657	−	0.978202i	$-0.433416\pi$
$41$	109.032			0.415314			0.207657	+	0.978202i	$0.433416\pi$
$42$		0			0
$43$			191.493i			0.679124i	0.940584	+	0.339562i	$0.110279\pi$
$43$			191.493i			0.679124i	−0.940584	+	0.339562i	$0.889721\pi$
$44$	261.132			0.894707
$45$		0			0
$46$	−198.622			−0.636634
$47$		−	414.420i		−	1.28616i	−0.765800	−	0.643078i	$-0.777657\pi$
$47$		−	414.420i		−	1.28616i	0.765800	−	0.643078i	$-0.222343\pi$
$48$		0			0
$49$	282.185			0.822697
$50$	−305.633	−	395.406i	−0.864461	−	1.11838i
$51$		0			0
$52$			526.796i			1.40487i
$53$		−	236.381i		−	0.612631i	−0.951930	−	0.306315i	$-0.900904\pi$
$53$		−	236.381i		−	0.612631i	0.951930	−	0.306315i	$-0.0990962\pi$
$54$		0			0
$55$	−118.147	−	346.039i	−0.289654	−	0.848362i
$56$	0.484192			0.00115541
$57$		0			0
$58$		−	1127.53i		−	2.55263i
$59$	784.024			1.73002			0.865011	−	0.501753i	$-0.167312\pi$
$59$	784.024			1.73002			0.865011	+	0.501753i	$0.167312\pi$
$60$		0			0
$61$	−535.138			−1.12324			−0.561618	−	0.827397i	$-0.689821\pi$
$61$	−535.138			−1.12324			−0.561618	+	0.827397i	$0.689821\pi$
$62$		−	590.363i		−	1.20929i
$63$		0			0
$64$	510.010			0.996114
$65$	698.084	−	238.345i	1.33210	−	0.454816i
$66$		0			0
$67$		−	121.069i		−	0.220761i	−0.993889	−	0.110380i	$-0.964793\pi$
$67$		−	121.069i		−	0.220761i	0.993889	−	0.110380i	$-0.0352069\pi$
$68$		−	124.441i		−	0.221921i
$69$		0			0
$70$	112.633	+	329.888i	0.192317	+	0.563273i
$71$	−590.432			−0.986921			−0.493460	−	0.869768i	$-0.664268\pi$
$71$	−590.432			−0.986921			−0.493460	+	0.869768i	$0.664268\pi$
$72$		0			0
$73$			986.078i			1.58098i	0.612473	+	0.790491i	$0.290175\pi$
$73$			986.078i			1.58098i	−0.612473	+	0.790491i	$0.709825\pi$
$74$	489.753			0.769360
$75$		0			0
$76$	412.526			0.622632
$77$			255.046i			0.377471i
$78$		0			0
$79$	607.025			0.864503			0.432251	−	0.901753i	$-0.357719\pi$
$79$	607.025			0.864503			0.432251	+	0.901753i	$0.357719\pi$
$80$	−231.649	−	678.472i	−0.323739	−	0.948194i
$81$		0			0
$82$			435.915i			0.587058i
$83$			1292.46i			1.70923i	0.519260	+	0.854616i	$0.326208\pi$
$83$			1292.46i			1.70923i	−0.519260	+	0.854616i	$0.673792\pi$
$84$		0			0
$85$	−164.903	+	56.3023i	−0.210426	+	0.0718452i
$86$	−765.598			−0.959961
$87$		0			0
$88$		−	2.03061i		−	0.00245981i
$89$	−1506.78			−1.79459			−0.897296	−	0.441428i	$-0.854472\pi$
$89$	−1506.78			−1.79459			−0.897296	+	0.441428i	$0.854472\pi$
$90$		0			0
$91$	−514.519			−0.592706
$92$		−	396.665i		−	0.449513i
$93$		0			0
$94$	1656.87			1.81802
$95$	−186.645	−	546.659i	−0.201572	−	0.590380i
$96$		0			0
$97$			610.887i			0.639445i	0.947511	+	0.319723i	$0.103590\pi$
$97$			610.887i			0.639445i	−0.947511	+	0.319723i	$0.896410\pi$
$98$			1128.19i			1.16290i
$99$		0			0
$100$	789.660	−	610.376i	0.789660	−	0.610376i
$101$	584.247			0.575592			0.287796	−	0.957692i	$-0.407077\pi$
$101$	584.247			0.575592			0.287796	+	0.957692i	$0.407077\pi$
$102$		0			0
$103$		−	1452.09i		−	1.38912i	−0.719437	−	0.694558i	$-0.755600\pi$
$103$		−	1452.09i		−	1.38912i	0.719437	−	0.694558i	$-0.244400\pi$
$104$	4.09645			0.00386241
$105$		0			0
$106$	945.065			0.865970
$107$		−	361.964i		−	0.327032i	−0.986541	−	0.163516i	$-0.947716\pi$
$107$		−	361.964i		−	0.327032i	0.986541	−	0.163516i	$-0.0522835\pi$
$108$		0			0
$109$	1073.79			0.943582			0.471791	−	0.881710i	$-0.343608\pi$
$109$	1073.79			0.943582			0.471791	+	0.881710i	$0.343608\pi$
$110$	1383.49	−	472.360i	1.19918	−	0.409434i
$111$		0			0
$112$			500.064i			0.421890i
$113$			1416.62i			1.17933i	0.807646	+	0.589667i	$0.200741\pi$
$113$			1416.62i			1.17933i	−0.807646	+	0.589667i	$0.799259\pi$
$114$		0			0
$115$	−525.641	+	179.468i	−0.426229	+	0.145526i
$116$	2251.78			1.80235
$117$		0			0
$118$			3134.57i			2.44543i
$119$	121.541			0.0936270
$120$		0			0
$121$	−261.384			−0.196382
$122$		−	2139.51i		−	1.58772i
$123$		0			0
$124$	1179.01			0.853854
$125$	−1166.12	−	770.258i	−0.834405	−	0.551152i
$126$		0			0
$127$		−	2709.85i		−	1.89339i	−0.322134	−	0.946694i	$-0.604400\pi$
$127$		−	2709.85i		−	1.89339i	0.322134	−	0.946694i	$-0.395600\pi$
$128$		−	7.94733i		−	0.00548790i
$129$		0			0
$130$	952.917	+	2790.98i	0.642895	+	1.88296i
$131$	621.192			0.414304			0.207152	−	0.978309i	$-0.433581\pi$
$131$	621.192			0.414304			0.207152	+	0.978309i	$0.433581\pi$
$132$		0			0
$133$			402.912i			0.262684i
$134$	484.042			0.312051
$135$		0			0
$136$	−0.967672			−0.000610127
$137$			1328.54i			0.828502i	0.910163	+	0.414251i	$0.135956\pi$
$137$			1328.54i			0.828502i	−0.910163	+	0.414251i	$0.864044\pi$
$138$		0			0
$139$	−2536.79			−1.54797			−0.773986	−	0.633203i	$-0.781740\pi$
$139$	−2536.79			−1.54797			−0.773986	+	0.633203i	$0.781740\pi$
$140$	−658.814	+	224.937i	−0.397714	+	0.135791i
$141$		0			0
$142$		−	2360.58i		−	1.39504i
$143$			2157.79i			1.26184i
$144$		0			0
$145$	−1018.80	−	2983.95i	−0.583497	−	1.70899i
$146$	−3942.40			−2.23476
$147$		0			0
$148$			978.079i			0.543227i
$149$	−321.699			−0.176877			−0.0884384	−	0.996082i	$-0.528188\pi$
$149$	−321.699			−0.176877			−0.0884384	+	0.996082i	$0.528188\pi$
$150$		0			0
$151$	−3196.36			−1.72262			−0.861311	−	0.508078i	$-0.830356\pi$
$151$	−3196.36			−1.72262			−0.861311	+	0.508078i	$0.830356\pi$
$152$		−	3.20787i		−	0.00171180i
$153$		0			0
$154$	−1019.69			−0.533565
$155$	−533.433	−	1562.36i	−0.276428	−	0.809626i
$156$		0			0
$157$			2607.69i			1.32558i	0.748804	+	0.662792i	$0.230628\pi$
$157$			2607.69i			1.32558i	−0.748804	+	0.662792i	$0.769372\pi$
$158$			2426.92i			1.22200i
$159$		0			0
$160$	2707.32	−	924.352i	1.33770	−	0.456728i
$161$	387.421			0.189646
$162$		0			0
$163$			642.287i			0.308637i	0.988021	+	0.154318i	$0.0493181\pi$
$163$			642.287i			0.308637i	−0.988021	+	0.154318i	$0.950682\pi$
$164$	−870.560			−0.414508
$165$		0			0
$166$	−5167.34			−2.41605
$167$			1999.64i			0.926569i	0.886209	+	0.463285i	$0.153329\pi$
$167$			1999.64i			0.926569i	−0.886209	+	0.463285i	$0.846671\pi$
$168$		0			0
$169$	−2156.03			−0.981353
$170$	−225.100	−	659.291i	−0.101555	−	0.297443i
$171$		0			0
$172$		−	1528.97i		−	0.677806i
$173$			3070.57i			1.34943i	0.738079	+	0.674714i	$0.235733\pi$
$173$			3070.57i			1.34943i	−0.738079	+	0.674714i	$0.764267\pi$
$174$		0			0
$175$	596.152	+	771.257i	0.257513	+	0.333152i
$176$	2097.17			0.898184
$177$		0			0
$178$		−	6024.21i		−	2.53671i
$179$	656.735			0.274227			0.137114	−	0.990555i	$-0.456217\pi$
$179$	656.735			0.274227			0.137114	+	0.990555i	$0.456217\pi$
$180$		0			0
$181$	−1504.70			−0.617919			−0.308960	−	0.951075i	$-0.599981\pi$
$181$	−1504.70			−0.617919			−0.308960	+	0.951075i	$0.599981\pi$
$182$		−	2057.08i		−	0.837806i
$183$		0			0
$184$	−3.08454			−0.00123584
$185$	1296.10	−	442.525i	0.515088	−	0.175865i
$186$		0			0
$187$		−	509.718i		−	0.199328i
$188$			3308.92i			1.28366i
$189$		0			0
$190$	2185.58	−	746.216i	0.834518	−	0.284927i
$191$	−4819.27			−1.82571			−0.912853	−	0.408288i	$-0.866126\pi$
$191$	−4819.27			−1.82571			−0.912853	+	0.408288i	$0.866126\pi$
$192$		0			0
$193$		−	4027.39i		−	1.50206i	−0.660267	−	0.751031i	$-0.729557\pi$
$193$		−	4027.39i		−	1.50206i	0.660267	−	0.751031i	$-0.270443\pi$
$194$	−2442.36			−0.903873
$195$		0			0
$196$	−2253.10			−0.821100
$197$			2132.04i			0.771075i	0.922692	+	0.385537i	$0.125984\pi$
$197$			2132.04i			0.771075i	−0.922692	+	0.385537i	$0.874016\pi$
$198$		0			0
$199$	−3024.96			−1.07756			−0.538779	−	0.842447i	$-0.681114\pi$
$199$	−3024.96			−1.07756			−0.538779	+	0.842447i	$0.681114\pi$
$200$	−4.74639	−	6.14053i	−0.00167810	−	0.00217100i
$201$		0			0
$202$			2335.86i			0.813615i
$203$			2199.31i			0.760399i
$204$		0			0
$205$	393.879	+	1153.62i	0.134194	+	0.393037i
$206$	5805.55			1.96355
$207$		0			0
$208$			4230.74i			1.41033i
$209$	1689.74			0.559242
$210$		0			0
$211$	−2191.00			−0.714856			−0.357428	−	0.933941i	$-0.616346\pi$
$211$	−2191.00			−0.714856			−0.357428	+	0.933941i	$0.616346\pi$
$212$			1887.38i			0.611442i
$213$		0			0
$214$	1447.15			0.462268
$215$	−2026.11	+	691.770i	−0.642696	+	0.219434i
$216$		0			0
$217$			1151.53i			0.360235i
$218$			4293.08i			1.33378i
$219$		0			0
$220$	943.344	+	2762.94i	0.289092	+	0.846716i
$221$	1028.28			0.312985
$222$		0			0
$223$			1159.78i			0.348271i	0.984722	+	0.174136i	$0.0557131\pi$
$223$			1159.78i			0.348271i	−0.984722	+	0.174136i	$0.944287\pi$
$224$	−1995.41			−0.595197
$225$		0			0
$226$	−5663.75			−1.66702
$227$			408.227i			0.119361i	0.998218	+	0.0596805i	$0.0190082\pi$
$227$			408.227i			0.119361i	−0.998218	+	0.0596805i	$0.980992\pi$
$228$		0			0
$229$	1858.12			0.536193			0.268097	−	0.963392i	$-0.413605\pi$
$229$	1858.12			0.536193			0.268097	+	0.963392i	$0.413605\pi$
$230$	−717.525	−	2101.54i	−0.205705	−	0.602486i
$231$		0			0
$232$		−	17.5103i		−	0.00495519i
$233$			2813.15i			0.790968i	0.918473	+	0.395484i	$0.129423\pi$
$233$			2813.15i			0.790968i	−0.918473	+	0.395484i	$0.870577\pi$
$234$		0			0
$235$	4384.82	−	1497.10i	1.21717	−	0.415574i
$236$	−6260.02			−1.72666
$237$		0			0
$238$			485.927i			0.132344i
$239$	−5756.48			−1.55797			−0.778987	−	0.627040i	$-0.784266\pi$
$239$	−5756.48			−1.55797			−0.778987	+	0.627040i	$0.784266\pi$
$240$		0			0
$241$	5740.04			1.53423			0.767114	−	0.641511i	$-0.221692\pi$
$241$	5740.04			1.53423			0.767114	+	0.641511i	$0.221692\pi$
$242$		−	1045.03i		−	0.277591i
$243$		0			0
$244$	4272.79			1.12105
$245$	1019.40	+	2985.69i	0.265824	+	0.778568i
$246$		0			0
$247$			3408.80i			0.878124i
$248$		−	9.16816i		−	0.00234750i
$249$		0			0
$250$	3079.54	−	4662.20i	0.779068	−	1.17945i
$251$	1120.86			0.281866			0.140933	−	0.990019i	$-0.454990\pi$
$251$	1120.86			0.281866			0.140933	+	0.990019i	$0.454990\pi$
$252$		0			0
$253$		−	1624.77i		−	0.403749i
$254$	10834.1			2.67636
$255$		0			0
$256$	4111.86			1.00387
$257$			982.609i			0.238496i	0.992864	+	0.119248i	$0.0380484\pi$
$257$			982.609i			0.238496i	−0.992864	+	0.119248i	$0.961952\pi$
$258$		0			0
$259$	−955.286			−0.229184
$260$	−5573.83	+	1903.06i	−1.32952	+	0.453933i
$261$		0			0
$262$			2483.56i			0.585630i
$263$		−	3419.85i		−	0.801813i	−0.916119	−	0.400907i	$-0.868695\pi$
$263$		−	3419.85i		−	0.801813i	0.916119	−	0.400907i	$-0.131305\pi$
$264$		0			0
$265$	2501.06	−	853.931i	0.579769	−	0.197949i
$266$	−1610.87			−0.371311
$267$		0			0
$268$			966.674i			0.220332i
$269$	−1425.68			−0.323142			−0.161571	−	0.986861i	$-0.551656\pi$
$269$	−1425.68			−0.323142			−0.161571	+	0.986861i	$0.551656\pi$
$270$		0			0
$271$	4234.80			0.949247			0.474624	−	0.880189i	$-0.342584\pi$
$271$	4234.80			0.949247			0.474624	+	0.880189i	$0.342584\pi$
$272$		−	999.394i		−	0.222784i
$273$		0			0
$274$	−5311.58			−1.17111
$275$	3234.50	−	2500.15i	0.709265	−	0.548235i
$276$		0			0
$277$			3258.65i			0.706835i	0.935466	+	0.353418i	$0.114981\pi$
$277$			3258.65i			0.706835i	−0.935466	+	0.353418i	$0.885019\pi$
$278$		−	10142.2i		−	2.18810i
$279$		0			0
$280$	1.74915	+	5.12305i	0.000373328	+	0.00109343i
$281$	−936.334			−0.198779			−0.0993897	−	0.995049i	$-0.531689\pi$
$281$	−936.334			−0.198779			−0.0993897	+	0.995049i	$0.531689\pi$
$282$		0			0
$283$			1368.62i			0.287477i	0.989616	+	0.143739i	$0.0459125\pi$
$283$			1368.62i			0.287477i	−0.989616	+	0.143739i	$0.954088\pi$
$284$	4714.29			0.985005
$285$		0			0
$286$	−8626.98			−1.78365
$287$		−	850.272i		−	0.174878i
$288$		0			0
$289$	4670.10			0.950559
$290$	11930.0	−	4073.24i	2.41571	−	0.824788i
$291$		0			0
$292$		−	7873.31i		−	1.57791i
$293$			1667.96i			0.332571i	0.986078	+	0.166285i	$0.0531773\pi$
$293$			1667.96i			0.332571i	−0.986078	+	0.166285i	$0.946823\pi$
$294$		0			0
$295$	2832.30	+	8295.47i	0.558993	+	1.63722i
$296$	7.60571			0.00149349
$297$		0			0
$298$		−	1286.17i		−	0.250020i
$299$	3277.73			0.633967
$300$		0			0
$301$	1493.34			0.285961
$302$		−	12779.2i		−	2.43497i
$303$		0			0
$304$	3313.03			0.625051
$305$	−1933.19	−	5662.09i	−0.362932	−	1.06299i
$306$		0			0
$307$		−	6894.86i		−	1.28179i	−0.767627	−	0.640897i	$-0.778563\pi$
$307$		−	6894.86i		−	1.28179i	0.767627	−	0.640897i	$-0.221437\pi$
$308$		−	2036.41i		−	0.376738i
$309$		0			0
$310$	6246.42	−	2132.70i	1.14443	−	0.390739i
$311$	−697.317			−0.127142			−0.0635711	−	0.997977i	$-0.520249\pi$
$311$	−697.317			−0.127142			−0.0635711	+	0.997977i	$0.520249\pi$
$312$		0			0
$313$			4889.18i			0.882917i	0.897282	+	0.441459i	$0.145539\pi$
$313$			4889.18i			0.882917i	−0.897282	+	0.441459i	$0.854461\pi$
$314$	−10425.7			−1.87375
$315$		0			0
$316$	−4846.78			−0.862824
$317$			5468.56i			0.968911i	0.874816	+	0.484455i	$0.160982\pi$
$317$			5468.56i			0.968911i	−0.874816	+	0.484455i	$0.839018\pi$
$318$		0			0
$319$	9223.47			1.61886
$320$	1842.42	+	5396.23i	0.321858	+	0.942683i
$321$		0			0
$322$			1548.93i			0.268070i
$323$		−	805.233i		−	0.138713i
$324$		0			0
$325$	5043.68	+	6525.14i	0.860840	+	1.11369i
$326$	−2567.90			−0.436266
$327$		0			0
$328$			6.76962i			0.00113960i
$329$	−3231.81			−0.541567
$330$		0			0
$331$	−5172.64			−0.858955			−0.429478	−	0.903078i	$-0.641302\pi$
$331$	−5172.64			−0.858955			−0.429478	+	0.903078i	$0.641302\pi$
$332$		−	10319.6i		−	1.70591i
$333$		0			0
$334$	−7994.69			−1.30973
$335$	1280.99	−	437.365i	0.208919	−	0.0713307i
$336$		0			0
$337$		−	409.004i		−	0.0661124i	−0.999453	−	0.0330562i	$-0.989476\pi$
$337$		−	409.004i		−	0.0661124i	0.999453	−	0.0330562i	$-0.0105240\pi$
$338$		−	8619.94i		−	1.38717i
$339$		0			0
$340$	1316.66	−	449.544i	0.210018	−	0.0717058i
$341$	4829.30			0.766925
$342$		0			0
$343$		−	4875.44i		−	0.767490i
$344$	−11.8895			−0.00186349
$345$		0			0
$346$	−12276.3			−1.90745
$347$			8027.94i			1.24197i	0.783824	+	0.620983i	$0.213266\pi$
$347$			8027.94i			1.24197i	−0.783824	+	0.620983i	$0.786734\pi$
$348$		0			0
$349$	4988.97			0.765196			0.382598	−	0.923915i	$-0.375029\pi$
$349$	4988.97			0.765196			0.382598	+	0.923915i	$0.375029\pi$
$350$	−3083.53	+	2383.45i	−0.470919	+	0.364002i
$351$		0			0
$352$			8368.38i			1.26715i
$353$		−	4238.90i		−	0.639133i	−0.947564	−	0.319567i	$-0.896463\pi$
$353$		−	4238.90i		−	0.639133i	0.947564	−	0.319567i	$-0.103537\pi$
$354$		0			0
$355$	−2132.95	−	6247.14i	−0.318887	−	0.933983i
$356$	12030.9			1.79111
$357$		0			0
$358$			2625.66i			0.387628i
$359$	9453.60			1.38981			0.694905	−	0.719101i	$-0.255446\pi$
$359$	9453.60			1.38981			0.694905	+	0.719101i	$0.255446\pi$
$360$		0			0
$361$	−4189.62			−0.610820
$362$		−	6015.88i		−	0.873446i
$363$		0			0
$364$	4108.16			0.591555
$365$	−10433.3	+	3562.23i	−1.49618	+	0.510837i
$366$		0			0
$367$		−	2750.63i		−	0.391231i	−0.980681	−	0.195616i	$-0.937330\pi$
$367$		−	2750.63i		−	0.391231i	0.980681	−	0.195616i	$-0.0626705\pi$
$368$		−	3185.65i		−	0.451260i
$369$		0			0
$370$	1769.24	+	5181.89i	0.248590	+	0.728091i
$371$	−1843.39			−0.257963
$372$		0			0
$373$		−	3417.55i		−	0.474407i	−0.971460	−	0.237204i	$-0.923769\pi$
$373$		−	3417.55i		−	0.474407i	0.971460	−	0.237204i	$-0.0762308\pi$
$374$	2037.88			0.281755
$375$		0			0
$376$	25.7308			0.00352916
$377$			18607.0i			2.54193i
$378$		0			0
$379$	9983.20			1.35304			0.676520	−	0.736424i	$-0.263487\pi$
$379$	9983.20			1.35304			0.676520	+	0.736424i	$0.263487\pi$
$380$	1490.26	+	4364.79i	0.201181	+	0.589234i
$381$		0			0
$382$		−	19267.7i		−	2.58069i
$383$		−	13456.7i		−	1.79531i	−0.440701	−	0.897654i	$-0.645270\pi$
$383$		−	13456.7i		−	1.79531i	0.440701	−	0.897654i	$-0.354730\pi$
$384$		0			0
$385$	−2698.55	+	921.360i	−0.357223	+	0.121966i
$386$	16101.7			2.12321
$387$		0			0
$388$		−	4877.61i		−	0.638204i
$389$	−8101.33			−1.05592			−0.527961	−	0.849269i	$-0.677043\pi$
$389$	−8101.33			−1.05592			−0.527961	+	0.849269i	$0.677043\pi$
$390$		0			0
$391$	−774.273			−0.100145
$392$			17.5205i			0.00225744i
$393$		0			0
$394$	−8524.03			−1.08994
$395$	2192.89	+	6422.71i	0.279332	+	0.818131i
$396$		0			0
$397$			7043.15i			0.890392i	0.895433	+	0.445196i	$0.146866\pi$
$397$			7043.15i			0.890392i	−0.895433	+	0.445196i	$0.853134\pi$
$398$		−	12094.0i		−	1.52316i
$399$		0			0
$400$	6341.83	−	4901.99i	0.792728	−	0.612748i
$401$	−2716.60			−0.338305			−0.169153	−	0.985590i	$-0.554103\pi$
$401$	−2716.60			−0.338305			−0.169153	+	0.985590i	$0.554103\pi$
$402$		0			0
$403$			9742.41i			1.20423i
$404$	−4664.91			−0.574475
$405$		0			0
$406$	−8792.96			−1.07485
$407$			4006.28i			0.487922i
$408$		0			0
$409$	6062.59			0.732948			0.366474	−	0.930428i	$-0.380565\pi$
$409$	6062.59			0.732948			0.366474	+	0.930428i	$0.380565\pi$
$410$	−4612.25	+	1574.75i	−0.555568	+	0.189686i
$411$		0			0
$412$			11594.2i			1.38642i
$413$		−	6114.13i		−	0.728467i
$414$		0			0
$415$	−13675.1	+	4669.05i	−1.61755	+	0.552276i
$416$	−16882.0			−1.98968
$417$		0			0
$418$			6755.67i			0.790504i
$419$	7008.83			0.817193			0.408596	−	0.912715i	$-0.366018\pi$
$419$	7008.83			0.817193			0.408596	+	0.912715i	$0.366018\pi$
$420$		0			0
$421$	−10812.3			−1.25168			−0.625842	−	0.779950i	$-0.715245\pi$
$421$	−10812.3			−1.25168			−0.625842	+	0.779950i	$0.715245\pi$
$422$		−	8759.74i		−	1.01047i
$423$		0			0
$424$	14.6766			0.00168103
$425$	−1191.43	−	1541.38i	−0.135983	−	0.175925i
$426$		0			0
$427$			4173.22i			0.472965i
$428$			2890.09i			0.326397i
$429$		0			0
$430$	−2765.74	−	8100.51i	−0.310176	−	0.908468i
$431$	8875.68			0.991941			0.495970	−	0.868339i	$-0.334812\pi$
$431$	8875.68			0.991941			0.495970	+	0.868339i	$0.334812\pi$
$432$		0			0
$433$		−	7039.08i		−	0.781239i	−0.920552	−	0.390619i	$-0.872261\pi$
$433$		−	7039.08i		−	0.781239i	0.920552	−	0.390619i	$-0.127739\pi$
$434$	−4603.89			−0.509202
$435$		0			0
$436$	−8573.65			−0.941751
$437$		−	2566.75i		−	0.280971i
$438$		0			0
$439$	5139.05			0.558710			0.279355	−	0.960188i	$-0.409879\pi$
$439$	5139.05			0.558710			0.279355	+	0.960188i	$0.409879\pi$
$440$	21.4851	−	7.33561i	0.00232787	−	0.000794798i
$441$		0			0
$442$			4111.13i			0.442413i
$443$		−	5074.43i		−	0.544229i	−0.962265	−	0.272115i	$-0.912277\pi$
$443$		−	5074.43i		−	0.544229i	0.962265	−	0.272115i	$-0.0877230\pi$
$444$		0			0
$445$	−5443.28	−	15942.7i	−0.579857	−	1.69833i
$446$	−4636.86			−0.492291
$447$		0			0
$448$		−	3977.26i		−	0.419437i
$449$	11783.0			1.23848			0.619238	−	0.785203i	$-0.287442\pi$
$449$	11783.0			1.23848			0.619238	+	0.785203i	$0.287442\pi$
$450$		0			0
$451$	−3565.88			−0.372307
$452$		−	11311.0i		−	1.17705i
$453$		0			0
$454$	−1632.11			−0.168720
$455$	−1858.71	−	5443.93i	−0.191511	−	0.560913i
$456$		0			0
$457$		−	11230.4i		−	1.14953i	−0.818320	−	0.574764i	$-0.805094\pi$
$457$		−	11230.4i		−	1.14953i	0.818320	−	0.574764i	$-0.194906\pi$
$458$			7428.88i			0.757923i
$459$		0			0
$460$	4196.97	−	1432.96i	0.425401	−	0.145244i
$461$	−7987.75			−0.806999			−0.403499	−	0.914980i	$-0.632206\pi$
$461$	−7987.75			−0.806999			−0.403499	+	0.914980i	$0.632206\pi$
$462$		0			0
$463$			16415.3i			1.64770i	0.566810	+	0.823848i	$0.308177\pi$
$463$			16415.3i			1.64770i	−0.566810	+	0.823848i	$0.691823\pi$
$464$	18084.3			1.80936
$465$		0			0
$466$	−11247.1			−1.11805
$467$		−	6002.60i		−	0.594790i	−0.954755	−	0.297395i	$-0.903882\pi$
$467$		−	6002.60i		−	0.594790i	0.954755	−	0.297395i	$-0.0961178\pi$
$468$		0			0
$469$	−944.146			−0.0929565
$470$	5985.49	+	17530.8i	0.587426	+	1.72050i
$471$		0			0
$472$			48.6790i			0.00474710i
$473$		−	6262.76i		−	0.608799i
$474$		0			0
$475$	5109.74	−	3949.63i	0.493581	−	0.381519i
$476$	−970.438			−0.0934453
$477$		0			0
$478$		−	23014.7i		−	2.20224i
$479$	−2516.64			−0.240059			−0.120029	−	0.992770i	$-0.538299\pi$
$479$	−2516.64			−0.240059			−0.120029	+	0.992770i	$0.538299\pi$
$480$		0			0
$481$	−8082.09			−0.766137
$482$			22949.0i			2.16867i
$483$		0			0
$484$	2087.01			0.196000
$485$	−6463.57	+	2206.84i	−0.605146	+	0.206613i
$486$		0			0
$487$		−	3893.26i		−	0.362260i	−0.983459	−	0.181130i	$-0.942025\pi$
$487$		−	3893.26i		−	0.362260i	0.983459	−	0.181130i	$-0.0579754\pi$
$488$		−	33.2260i		−	0.00308211i
$489$		0			0
$490$	−11937.0	+	4075.61i	−1.10053	+	0.375750i
$491$	−8911.50			−0.819084			−0.409542	−	0.912291i	$-0.634312\pi$
$491$	−8911.50			−0.819084			−0.409542	+	0.912291i	$0.634312\pi$
$492$		0			0
$493$		−	4395.38i		−	0.401538i
$494$	−13628.6			−1.24125
$495$		0			0
$496$	9468.71			0.857173
$497$			4604.42i			0.415567i
$498$		0			0
$499$	−8639.01			−0.775021			−0.387510	−	0.921865i	$-0.626665\pi$
$499$	−8639.01			−0.775021			−0.387510	+	0.921865i	$0.626665\pi$
$500$	9310.82	+	6150.10i	0.832785	+	0.550082i
$501$		0			0
$502$			4481.28i			0.398425i
$503$		−	12847.4i		−	1.13884i	−0.822046	−	0.569422i	$-0.807167\pi$
$503$		−	12847.4i		−	1.13884i	0.822046	−	0.569422i	$-0.192833\pi$
$504$		0			0
$505$	2110.60	+	6181.71i	0.185981	+	0.544717i
$506$	6495.92			0.570709
$507$		0			0
$508$			21636.7i			1.88971i
$509$	−3956.77			−0.344560			−0.172280	−	0.985048i	$-0.555113\pi$
$509$	−3956.77			−0.344560			−0.172280	+	0.985048i	$0.555113\pi$
$510$		0			0
$511$	7689.83			0.665710
$512$			16375.9i			1.41351i
$513$		0			0
$514$	−3928.53			−0.337121
$515$	15364.1	−	5245.71i	1.31460	−	0.448842i
$516$		0			0
$517$			13553.6i			1.15297i
$518$		−	3819.29i		−	0.323957i
$519$		0			0
$520$	14.7985	+	43.3431i	0.00124800	+	0.00365523i
$521$	−6392.00			−0.537502			−0.268751	−	0.963210i	$-0.586611\pi$
$521$	−6392.00			−0.537502			−0.268751	+	0.963210i	$0.586611\pi$
$522$		0			0
$523$		−	13029.6i		−	1.08938i	−0.838638	−	0.544690i	$-0.816647\pi$
$523$		−	13029.6i		−	1.08938i	0.838638	−	0.544690i	$-0.183353\pi$
$524$	−4959.89			−0.413500
$525$		0			0
$526$	13672.8			1.13338
$527$		−	2301.37i		−	0.190226i
$528$		0			0
$529$	9698.94			0.797151
$530$	3414.06	+	9999.38i	0.279807	+	0.819520i
$531$		0			0
$532$		−	3217.04i		−	0.262174i
$533$		−	7193.64i		−	0.584598i
$534$		0			0
$535$	3829.81	−	1307.60i	0.309490	−	0.105668i
$536$	7.51702			0.000605757
$537$		0			0
$538$		−	5699.95i		−	0.456770i
$539$	−9228.86			−0.737505
$540$		0			0
$541$	−9481.72			−0.753514			−0.376757	−	0.926312i	$-0.622961\pi$
$541$	−9481.72			−0.753514			−0.376757	+	0.926312i	$0.622961\pi$
$542$			16931.0i			1.34179i
$543$		0			0
$544$	3987.90			0.314301
$545$	3879.09	+	11361.4i	0.304884	+	0.892969i
$546$		0			0
$547$		−	22132.1i		−	1.72998i	−0.501788	−	0.864991i	$-0.667324\pi$
$547$		−	22132.1i		−	1.72998i	0.501788	−	0.864991i	$-0.332676\pi$
$548$		−	10607.7i		−	0.826893i
$549$		0			0
$550$	9995.73	+	12931.7i	0.774944	+	1.00257i
$551$	14570.9			1.12657
$552$		0			0
$553$		−	4733.83i		−	0.364019i
$554$	−13028.3			−0.999131
$555$		0			0
$556$	20255.0			1.54497
$557$			4426.25i			0.336708i	0.985727	+	0.168354i	$0.0538451\pi$
$557$			4426.25i			0.336708i	−0.985727	+	0.168354i	$0.946155\pi$
$558$		0			0
$559$	12634.2			0.955939
$560$	−5291.00	+	1806.49i	−0.399260	+	0.136318i
$561$		0			0
$562$		−	3743.52i		−	0.280980i
$563$		−	3173.26i		−	0.237543i	−0.992922	−	0.118772i	$-0.962104\pi$
$563$		−	3173.26i		−	0.237543i	0.992922	−	0.118772i	$-0.0378957\pi$
$564$		0			0
$565$	−14988.8	+	5117.58i	−1.11608	+	0.381059i
$566$	−5471.83			−0.406357
$567$		0			0
$568$		−	36.6591i		−	0.00270807i
$569$	3665.79			0.270084			0.135042	−	0.990840i	$-0.456883\pi$
$569$	3665.79			0.270084			0.135042	+	0.990840i	$0.456883\pi$
$570$		0			0
$571$	9022.74			0.661279			0.330639	−	0.943757i	$-0.392736\pi$
$571$	9022.74			0.661279			0.330639	+	0.943757i	$0.392736\pi$
$572$		−	17228.8i		−	1.25940i
$573$		0			0
$574$	3399.44			0.247195
$575$	−3797.78	−	4913.28i	−0.275440	−	0.356344i
$576$		0			0
$577$			16600.1i			1.19769i	0.800863	+	0.598847i	$0.204374\pi$
$577$			16600.1i			1.19769i	−0.800863	+	0.598847i	$0.795626\pi$
$578$			18671.3i			1.34364i
$579$		0			0
$580$	8134.61	+	23825.3i	0.582364	+	1.70567i
$581$	10079.1			0.719713
$582$		0			0
$583$			7730.84i			0.549191i
$584$	−61.2242			−0.00433815
$585$		0			0
$586$	−6668.60			−0.470098
$587$			18193.9i			1.27929i	0.768671	+	0.639645i	$0.220918\pi$
$587$			18193.9i			1.27929i	−0.768671	+	0.639645i	$0.779082\pi$
$588$		0			0
$589$	7629.14			0.533707
$590$	−33165.8	+	11323.7i	−2.31426	+	0.790152i
$591$		0			0
$592$			7855.04i			0.545338i
$593$			448.648i			0.0310688i	0.999879	+	0.0155344i	$0.00494495\pi$
$593$			448.648i			0.0310688i	−0.999879	+	0.0155344i	$0.995055\pi$
$594$		0			0
$595$	439.068	+	1285.98i	0.0302521	+	0.0886049i
$596$	2568.60			0.176533
$597$		0			0
$598$			13104.6i			0.896130i
$599$	−8787.84			−0.599435			−0.299717	−	0.954028i	$-0.596892\pi$
$599$	−8787.84			−0.599435			−0.299717	+	0.954028i	$0.596892\pi$
$600$		0			0
$601$	9787.87			0.664319			0.332159	−	0.943223i	$-0.392223\pi$
$601$	9787.87			0.664319			0.332159	+	0.943223i	$0.392223\pi$
$602$			5970.44i			0.404214i
$603$		0			0
$604$	25521.2			1.71928
$605$	−944.254	−	2765.61i	−0.0634535	−	0.185848i
$606$		0			0
$607$			12769.7i			0.853883i	0.904279	+	0.426942i	$0.140409\pi$
$607$			12769.7i			0.853883i	−0.904279	+	0.426942i	$0.859591\pi$
$608$			13220.0i			0.881815i
$609$		0			0
$610$	22637.4	−	7729.02i	1.50256	−	0.513015i
$611$	−27342.4			−1.81040
$612$		0			0
$613$			22910.6i			1.50954i	0.655988	+	0.754771i	$0.272252\pi$
$613$			22910.6i			1.50954i	−0.655988	+	0.754771i	$0.727748\pi$
$614$	27566.1			1.81185
$615$		0			0
$616$	−15.8355			−0.00103576
$617$			22841.4i			1.49038i	0.666854	+	0.745188i	$0.267640\pi$
$617$			22841.4i			1.49038i	−0.666854	+	0.745188i	$0.732360\pi$
$618$		0			0
$619$	495.421			0.0321691			0.0160845	−	0.999871i	$-0.494880\pi$
$619$	495.421			0.0321691			0.0160845	+	0.999871i	$0.494880\pi$
$620$	4259.18	+	12474.6i	0.275892	+	0.808054i
$621$		0			0
$622$		−	2787.92i		−	0.179719i
$623$			11750.5i			0.755656i
$624$		0			0
$625$	3937.19	−	15120.8i	0.251980	−	0.967732i
$626$	−19547.2			−1.24803
$627$		0			0
$628$		−	20821.0i		−	1.32301i
$629$	1909.17			0.121023
$630$		0			0
$631$	25481.5			1.60761			0.803807	−	0.594890i	$-0.202805\pi$
$631$	25481.5			1.60761			0.803807	+	0.594890i	$0.202805\pi$
$632$			37.6894i			0.00237216i
$633$		0			0
$634$	−21863.6			−1.36958
$635$	28671.9	−	9789.38i	1.79183	−	0.611779i
$636$		0			0
$637$		−	18617.9i		−	1.15803i
$638$			36876.0i			2.28830i
$639$		0			0
$640$	84.0877	−	28.7099i	0.00519353	−	0.00177321i
$641$	2913.25			0.179511			0.0897555	−	0.995964i	$-0.471391\pi$
$641$	2913.25			0.179511			0.0897555	+	0.995964i	$0.471391\pi$
$642$		0			0
$643$			15121.2i			0.927404i	0.885991	+	0.463702i	$0.153479\pi$
$643$			15121.2i			0.927404i	−0.885991	+	0.463702i	$0.846521\pi$
$644$	−3093.35			−0.189278
$645$		0			0
$646$	3219.37			0.196075
$647$			8436.06i			0.512605i	0.966597	+	0.256303i	$0.0825044\pi$
$647$			8436.06i			0.512605i	−0.966597	+	0.256303i	$0.917496\pi$
$648$		0			0
$649$	−25641.5			−1.55087
$650$	−26087.9	+	20164.9i	−1.57423	+	1.21682i
$651$		0			0
$652$		−	5128.32i		−	0.308037i
$653$			17146.6i			1.02756i	0.857922	+	0.513780i	$0.171755\pi$
$653$			17146.6i			1.02756i	−0.857922	+	0.513780i	$0.828245\pi$
$654$		0			0
$655$	2244.07	+	6572.60i	0.133867	+	0.392081i
$656$	−6991.54			−0.416119
$657$		0			0
$658$		−	12921.0i		−	0.765519i
$659$	1774.60			0.104900			0.0524498	−	0.998624i	$-0.483297\pi$
$659$	1774.60			0.104900			0.0524498	+	0.998624i	$0.483297\pi$
$660$		0			0
$661$	−7159.71			−0.421302			−0.210651	−	0.977561i	$-0.567558\pi$
$661$	−7159.71			−0.421302			−0.210651	+	0.977561i	$0.567558\pi$
$662$		−	20680.5i		−	1.21416i
$663$		0			0
$664$	−80.2473			−0.00469006
$665$	−4263.07	+	1455.53i	−0.248593	+	0.0848767i
$666$		0			0
$667$		−	14010.6i		−	0.813335i
$668$		−	15966.1i		−	0.924771i
$669$		0			0
$670$	1748.61	+	5121.47i	0.100828	+	0.295313i
$671$	17501.7			1.00692
$672$		0			0
$673$		−	15556.7i		−	0.891034i	−0.895273	−	0.445517i	$-0.853020\pi$
$673$		−	15556.7i		−	0.891034i	0.895273	−	0.445517i	$-0.146980\pi$
$674$	1635.22			0.0934516
$675$		0			0
$676$	17214.8			0.979448
$677$			4461.98i			0.253306i	0.991947	+	0.126653i	$0.0404234\pi$
$677$			4461.98i			0.253306i	−0.991947	+	0.126653i	$0.959577\pi$
$678$		0			0
$679$	4763.94			0.269254
$680$	−3.49573	−	10.2386i	−0.000197140	−	0.000577400i
$681$		0			0
$682$			19307.8i			1.08407i
$683$		−	33470.5i		−	1.87513i	−0.347808	−	0.937566i	$-0.613074\pi$
$683$		−	33470.5i		−	1.87513i	0.347808	−	0.937566i	$-0.386926\pi$
$684$		0			0
$685$	−14056.8	+	4799.37i	−0.784061	+	0.267700i
$686$	19492.3			1.08487
$687$		0			0
$688$		−	12279.3i		−	0.680440i
$689$	−15595.8			−0.862342
$690$		0			0
$691$	−20280.9			−1.11653			−0.558264	−	0.829664i	$-0.688532\pi$
$691$	−20280.9			−1.11653			−0.558264	+	0.829664i	$0.688532\pi$
$692$		−	24516.9i		−	1.34681i
$693$		0			0
$694$	−32096.2			−1.75555
$695$	−9164.21	−	26840.9i	−0.500170	−	1.46494i
$696$		0			0
$697$			1699.29i			0.0923463i
$698$			19946.2i			1.08163i
$699$		0			0
$700$	−4759.96	−	6158.08i	−0.257013	−	0.332505i
$701$	−3762.36			−0.202714			−0.101357	−	0.994850i	$-0.532318\pi$
$701$	−3762.36			−0.202714			−0.101357	+	0.994850i	$0.532318\pi$
$702$		0			0
$703$			6328.97i			0.339547i
$704$	−16679.9			−0.892964
$705$		0			0
$706$	16947.4			0.903432
$707$		−	4556.19i		−	0.242367i
$708$		0			0
$709$	−1643.51			−0.0870570			−0.0435285	−	0.999052i	$-0.513860\pi$
$709$	−1643.51			−0.0870570			−0.0435285	+	0.999052i	$0.513860\pi$
$710$	24976.4	−	8527.64i	1.32021	−	0.450756i
$711$		0			0
$712$		−	93.5541i		−	0.00492428i
$713$		−	7335.81i		−	0.385313i
$714$		0			0
$715$	−22830.8	+	7795.07i	−1.19416	+	0.407719i
$716$	−5243.68			−0.273695
$717$		0			0
$718$			37796.0i			1.96453i
$719$	3748.68			0.194440			0.0972200	−	0.995263i	$-0.469005\pi$
$719$	3748.68			0.194440			0.0972200	+	0.995263i	$0.469005\pi$
$720$		0			0
$721$	−11324.0			−0.584921
$722$		−	16750.3i		−	0.863411i
$723$		0			0
$724$	12014.2			0.616720
$725$	27891.7	−	21559.2i	1.42879	−	1.10440i
$726$		0			0
$727$		−	12221.3i		−	0.623470i	−0.950169	−	0.311735i	$-0.899090\pi$
$727$		−	12221.3i		−	0.623470i	0.950169	−	0.311735i	$-0.100910\pi$
$728$		−	31.9458i		−	0.00162636i
$729$		0			0
$730$	−14242.0	−	41713.1i	−0.722081	−	2.11489i
$731$	−2984.48			−0.151005
$732$		0			0
$733$		−	31862.7i		−	1.60556i	−0.596274	−	0.802781i	$-0.703353\pi$
$733$		−	31862.7i		−	1.60556i	0.596274	−	0.802781i	$-0.296647\pi$
$734$	10997.2			0.553016
$735$		0			0
$736$	12711.7			0.636632
$737$			3959.57i			0.197900i
$738$		0			0
$739$	−9905.64			−0.493078			−0.246539	−	0.969133i	$-0.579293\pi$
$739$	−9905.64			−0.493078			−0.246539	+	0.969133i	$0.579293\pi$
$740$	−10348.7	+	3533.33i	−0.514088	+	0.175524i
$741$		0			0
$742$		−	7369.99i		−	0.364637i
$743$		−	29282.9i		−	1.44588i	−0.690913	−	0.722938i	$-0.742791\pi$
$743$		−	29282.9i		−	1.44588i	0.690913	−	0.722938i	$-0.257209\pi$
$744$		0			0
$745$	−1162.14	−	3403.78i	−0.0571512	−	0.167389i
$746$	13663.6			0.670587
$747$		0			0
$748$			4069.83i			0.198941i
$749$	−2822.74			−0.137705
$750$		0			0
$751$	−26361.7			−1.28090			−0.640448	−	0.768001i	$-0.721251\pi$
$751$	−26361.7			−1.28090			−0.640448	+	0.768001i	$0.721251\pi$
$752$			26574.3i			1.28865i
$753$		0			0
$754$	−74391.9			−3.59309
$755$	−11546.9	−	33819.5i	−0.556602	−	1.63022i
$756$		0			0
$757$		−	5667.66i		−	0.272120i	−0.990701	−	0.136060i	$-0.956556\pi$
$757$		−	5667.66i		−	0.272120i	0.990701	−	0.136060i	$-0.0434439\pi$
$758$			39913.4i			1.91256i
$759$		0			0
$760$	33.9413	−	11.5885i	0.00161998	−	0.000553104i
$761$	36532.3			1.74020			0.870101	−	0.492874i	$-0.164054\pi$
$761$	36532.3			1.74020			0.870101	+	0.492874i	$0.164054\pi$
$762$		0			0
$763$		−	8373.85i		−	0.397318i
$764$	38479.3			1.82216
$765$		0			0
$766$	53800.5			2.53772
$767$		−	51728.0i		−	2.43519i
$768$		0			0
$769$	−40900.2			−1.91795			−0.958973	−	0.283498i	$-0.908505\pi$
$769$	−40900.2			−1.91795			−0.958973	+	0.283498i	$0.908505\pi$
$770$	−3683.65	−	10789.0i	−0.172402	−	0.504945i
$771$		0			0
$772$			32156.6i			1.49915i
$773$			28081.8i			1.30664i	0.757083	+	0.653319i	$0.226624\pi$
$773$			28081.8i			1.30664i	−0.757083	+	0.653319i	$0.773376\pi$
$774$		0			0
$775$	14603.7	−	11288.1i	0.676880	−	0.523202i
$776$	−37.9292			−0.00175461
$777$		0			0
$778$		−	32389.6i		−	1.49257i
$779$	−5633.23			−0.259091
$780$		0			0
$781$	19310.1			0.884723
$782$		−	3095.59i		−	0.141558i
$783$		0			0
$784$	−18094.8			−0.824291
$785$	−27591.0	+	9420.34i	−1.25448	+	0.428314i
$786$		0			0
$787$			41047.8i			1.85921i	0.368558	+	0.929605i	$0.379852\pi$
$787$			41047.8i			1.85921i	−0.368558	+	0.929605i	$0.620148\pi$
$788$		−	17023.2i		−	0.769578i
$789$		0			0
$790$	−25678.4	+	8767.30i	−1.15645	+	0.394844i
$791$	11047.4			0.496587
$792$		0			0
$793$			35307.0i			1.58107i
$794$	−28158.9			−1.25859
$795$		0			0
$796$	24152.7			1.07547
$797$			3768.31i			0.167478i	0.996488	+	0.0837392i	$0.0266863\pi$
$797$			3768.31i			0.167478i	−0.996488	+	0.0837392i	$0.973314\pi$
$798$		0			0
$799$	6458.88			0.285981
$800$	19560.5	+	25305.9i	0.864458	+	1.11837i
$801$		0			0
$802$		−	10861.1i		−	0.478203i
$803$		−	32249.7i		−	1.41727i
$804$		0			0
$805$	1399.57	+	4099.16i	0.0612773	+	0.179474i
$806$	−38950.7			−1.70221
$807$		0			0
$808$			36.2751i			0.00157940i
$809$	24285.5			1.05542			0.527708	−	0.849426i	$-0.323051\pi$
$809$	24285.5			1.05542			0.527708	+	0.849426i	$0.323051\pi$
$810$		0			0
$811$	−42698.1			−1.84875			−0.924374	−	0.381487i	$-0.875412\pi$
$811$	−42698.1			−1.84875			−0.924374	+	0.381487i	$0.875412\pi$
$812$		−	17560.3i		−	0.758923i
$813$		0			0
$814$	−16017.4			−0.689691
$815$	−6795.80	+	2320.27i	−0.292081	+	0.0997246i
$816$		0			0
$817$		−	9893.67i		−	0.423667i
$818$			24238.6i			1.03604i
$819$		0			0
$820$	−3144.91	−	9211.07i	−0.133933	−	0.392274i
$821$	9160.55			0.389410			0.194705	−	0.980862i	$-0.437625\pi$
$821$	9160.55			0.389410			0.194705	+	0.980862i	$0.437625\pi$
$822$		0			0
$823$		−	37097.7i		−	1.57126i	−0.618697	−	0.785629i	$-0.712339\pi$
$823$		−	37097.7i		−	1.57126i	0.618697	−	0.785629i	$-0.287661\pi$
$824$	90.1585			0.00381167
$825$		0			0
$826$	24444.7			1.02971
$827$		−	24878.9i		−	1.04610i	−0.852302	−	0.523050i	$-0.824794\pi$
$827$		−	24878.9i		−	1.04610i	0.852302	−	0.523050i	$-0.175206\pi$
$828$		0			0
$829$	8788.86			0.368214			0.184107	−	0.982906i	$-0.441061\pi$
$829$	8788.86			0.368214			0.184107	+	0.982906i	$0.441061\pi$
$830$	−18667.1	−	54673.8i	−0.780657	−	2.28645i
$831$		0			0
$832$		−	33649.2i		−	1.40214i
$833$			4397.95i			0.182929i
$834$		0			0
$835$	−21157.5	+	7223.75i	−0.876868	+	0.299387i
$836$	−13491.7			−0.558157
$837$		0			0
$838$			28021.7i			1.15512i
$839$	6041.34			0.248594			0.124297	−	0.992245i	$-0.460332\pi$
$839$	6041.34			0.248594			0.124297	+	0.992245i	$0.460332\pi$
$840$		0			0
$841$	55146.5			2.26112
$842$		−	43228.2i		−	1.76929i
$843$		0			0
$844$	17494.0			0.713468
$845$	−7788.71	−	22812.2i	−0.317088	−	0.928714i
$846$		0			0
$847$			2038.38i			0.0826912i
$848$			15157.7i			0.613818i
$849$		0			0
$850$	6162.53	−	4763.40i	0.248674	−	0.192216i
$851$	6085.63			0.245138
$852$		0			0
$853$		−	19541.8i		−	0.784405i	−0.919879	−	0.392203i	$-0.871713\pi$
$853$		−	19541.8i		−	0.784405i	0.919879	−	0.392203i	$-0.128287\pi$
$854$	−16684.8			−0.668549
$855$		0			0
$856$	22.4739			0.000897361
$857$			34985.7i			1.39450i	0.716826	+	0.697252i	$0.245594\pi$
$857$			34985.7i			1.39450i	−0.716826	+	0.697252i	$0.754406\pi$
$858$		0			0
$859$	3118.04			0.123849			0.0619244	−	0.998081i	$-0.480276\pi$
$859$	3118.04			0.123849			0.0619244	+	0.998081i	$0.480276\pi$
$860$	16177.4	−	5523.42i	0.641449	−	0.219008i
$861$		0			0
$862$			35485.5i			1.40214i
$863$			8308.05i			0.327705i	0.986485	+	0.163852i	$0.0523921\pi$
$863$			8308.05i			0.327705i	−0.986485	+	0.163852i	$0.947608\pi$
$864$		0			0
$865$	−32488.5	+	11092.5i	−1.27704	+	0.436018i
$866$	28142.6			1.10430
$867$		0			0
$868$		−	9194.36i		−	0.359536i
$869$	−19852.8			−0.774981
$870$		0			0
$871$	−7987.85			−0.310744
$872$			66.6702i			0.00258915i
$873$		0			0
$874$	10262.0			0.397160
$875$	−6006.78	+	9093.84i	−0.232076	+	0.351346i
$876$		0			0
$877$		−	31065.2i		−	1.19612i	−0.801452	−	0.598059i	$-0.795939\pi$
$877$		−	31065.2i		−	1.19612i	0.801452	−	0.598059i	$-0.204061\pi$
$878$			20546.2i			0.789751i
$879$		0			0
$880$	7576.08	+	22189.4i	0.290215	+	0.850006i
$881$	−27176.4			−1.03927			−0.519634	−	0.854389i	$-0.673932\pi$
$881$	−27176.4			−1.03927			−0.519634	+	0.854389i	$0.673932\pi$
$882$		0			0
$883$			17811.1i			0.678813i	0.940640	+	0.339406i	$0.110226\pi$
$883$			17811.1i			0.678813i	−0.940640	+	0.339406i	$0.889774\pi$
$884$	−8210.29			−0.312378
$885$		0			0
$886$	20287.9			0.769283
$887$		−	9708.93i		−	0.367524i	−0.982971	−	0.183762i	$-0.941172\pi$
$887$		−	9708.93i		−	0.367524i	0.982971	−	0.183762i	$-0.0588276\pi$
$888$		0			0
$889$	−21132.5			−0.797256
$890$	63739.9	−	21762.6i	2.40064	−	0.819644i
$891$		0			0
$892$		−	9260.22i		−	0.347595i
$893$			21411.4i			0.802359i
$894$		0			0
$895$	2372.47	+	6948.67i	0.0886065	+	0.259518i
$896$	−61.9764			−0.00231081
$897$		0			0
$898$			47109.2i			1.75062i
$899$	41643.8			1.54494
$900$		0			0
$901$	3684.08			0.136220
$902$		−	14256.6i		−	0.526266i
$903$		0			0
$904$	−87.9562			−0.00323604
$905$	−5435.75	−	15920.7i	−0.199658	−	0.584774i
$906$		0			0
$907$			1793.35i			0.0656530i	0.999461	+	0.0328265i	$0.0104509\pi$
$907$			1793.35i			0.0656530i	−0.999461	+	0.0328265i	$0.989549\pi$
$908$		−	3259.47i		−	0.119129i
$909$		0			0
$910$	21765.2	−	7431.23i	0.792866	−	0.270706i
$911$	40163.5			1.46068			0.730338	−	0.683086i	$-0.239362\pi$
$911$	40163.5			1.46068			0.730338	+	0.683086i	$0.239362\pi$
$912$		0			0
$913$		−	42270.0i		−	1.53224i
$914$	44899.6			1.62489
$915$		0			0
$916$	−14836.1			−0.535152
$917$		−	4844.30i		−	0.174453i
$918$		0			0
$919$	19871.3			0.713268			0.356634	−	0.934244i	$-0.383924\pi$
$919$	19871.3			0.713268			0.356634	+	0.934244i	$0.383924\pi$
$920$	−11.1429	−	32.6363i	−0.000399317	−	0.00116955i
$921$		0			0
$922$		−	31935.5i		−	1.14071i
$923$			38955.2i			1.38920i
$924$		0			0
$925$	9364.39	+	12115.0i	0.332864	+	0.430635i
$926$	−65629.3			−2.32906
$927$		0			0
$928$			72161.9i			2.55262i
$929$	3690.27			0.130327			0.0651636	−	0.997875i	$-0.479243\pi$
$929$	3690.27			0.130327			0.0651636	+	0.997875i	$0.479243\pi$
$930$		0			0
$931$	−14579.4			−0.513233
$932$		−	22461.5i		−	0.789432i
$933$		0			0
$934$	23998.7			0.840752
$935$	5393.14	−	1841.37i	0.188636	−	0.0644055i
$936$		0			0
$937$			15191.5i			0.529653i	0.964296	+	0.264826i	$0.0853147\pi$
$937$			15191.5i			0.529653i	−0.964296	+	0.264826i	$0.914685\pi$
$938$		−	3774.75i		−	0.131397i
$939$		0			0
$940$	−35010.5	+	11953.5i	−1.21480	+	0.414768i
$941$	−29396.9			−1.01840			−0.509198	−	0.860650i	$-0.670058\pi$
$941$	−29396.9			−1.01840			−0.509198	+	0.860650i	$0.670058\pi$
$942$		0			0
$943$			5416.64i			0.187052i
$944$	−50274.8			−1.73337
$945$		0			0
$946$	25038.9			0.860554
$947$			1602.80i			0.0549991i	0.999622	+	0.0274995i	$0.00875448\pi$
$947$			1602.80i			0.0549991i	−0.999622	+	0.0274995i	$0.991246\pi$
$948$		0			0
$949$	65059.0			2.22540
$950$	15790.9	+	20429.1i	0.539288	+	0.697691i
$951$		0			0
$952$			7.54629i			0.000256908i
$953$			14868.1i			0.505377i	0.967548	+	0.252689i	$0.0813148\pi$
$953$			14868.1i			0.505377i	−0.967548	+	0.252689i	$0.918685\pi$
$954$		0			0
$955$	−17409.7	−	50990.9i	−0.589910	−	1.72778i
$956$	45962.4			1.55495
$957$		0			0
$958$		−	10061.7i		−	0.339329i
$959$	10360.5			0.348860
$960$		0			0
$961$	−7986.78			−0.268094
$962$		−	32312.7i		−	1.08295i
$963$		0			0
$964$	−45831.2			−1.53125
$965$	42612.3	−	14549.0i	1.42149	−	0.485336i
$966$		0			0
$967$			44013.7i			1.46369i	0.681474	+	0.731843i	$0.261339\pi$
$967$			44013.7i			1.46369i	−0.681474	+	0.731843i	$0.738661\pi$
$968$		−	16.2290i		−	0.000538862i
$969$		0			0
$970$	−8823.08	−	25841.7i	−0.292054	−	0.855390i
$971$	−50954.6			−1.68405			−0.842024	−	0.539440i	$-0.818636\pi$
$971$	−50954.6			−1.68405			−0.842024	+	0.539440i	$0.818636\pi$
$972$		0			0
$973$			19782.9i			0.651810i
$974$	15565.5			0.512064
$975$		0			0
$976$	34315.2			1.12541
$977$		−	18626.3i		−	0.609935i	−0.952363	−	0.304968i	$-0.901354\pi$
$977$		−	18626.3i		−	0.609935i	0.952363	−	0.304968i	$-0.0986456\pi$
$978$		0			0
$979$	49279.3			1.60876
$980$	−8139.36	−	23839.2i	−0.265308	−	0.777056i
$981$		0			0
$982$		−	35628.7i		−	1.15780i
$983$			11506.9i			0.373361i	0.982421	+	0.186681i	$0.0597730\pi$
$983$			11506.9i			0.373361i	−0.982421	+	0.186681i	$0.940227\pi$
$984$		0			0
$985$	−22558.3	+	7702.04i	−0.729714	+	0.249145i
$986$	17573.0			0.567584
$987$		0			0
$988$		−	27217.5i		−	0.876420i
$989$	−9513.27			−0.305869
$990$		0			0
$991$	31332.2			1.00434			0.502170	−	0.864769i	$-0.332535\pi$
$991$	31332.2			1.00434			0.502170	+	0.864769i	$0.332535\pi$
$992$			37783.1i			1.20929i
$993$		0			0
$994$	−18408.8			−0.587415
$995$	−10927.7	−	32006.0i	−0.348173	−	1.01976i
$996$		0			0
$997$		−	29027.6i		−	0.922080i	−0.887379	−	0.461040i	$-0.847476\pi$
$997$		−	29027.6i		−	0.922080i	0.887379	−	0.461040i	$-0.152524\pi$
$998$		−	34539.3i		−	1.09551i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.b.244.7	yes	8
3.2	odd	2		405.4.b.c.244.2	yes	8
5.2	odd	4		2025.4.a.bc.1.2		8
5.3	odd	4		2025.4.a.bc.1.7		8
5.4	even	2	inner	405.4.b.b.244.2	✓	8
15.2	even	4		2025.4.a.bd.1.7		8
15.8	even	4		2025.4.a.bd.1.2		8
15.14	odd	2		405.4.b.c.244.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.2	✓	8	5.4	even	2	inner
405.4.b.b.244.7	yes	8	1.1	even	1	trivial
405.4.b.c.244.2	yes	8	3.2	odd	2
405.4.b.c.244.7	yes	8	15.14	odd	2
2025.4.a.bc.1.2		8	5.2	odd	4
2025.4.a.bc.1.7		8	5.3	odd	4
2025.4.a.bd.1.2		8	15.8	even	4
2025.4.a.bd.1.7		8	15.2	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.99806i q^{2} -7.98447 q^{4} +(3.61252 + 10.5806i) q^{5} -7.79840i q^{7} +0.0620886i q^{8} +O(q^{10})\)	\(q+3.99806i q^{2} -7.98447 q^{4} +(3.61252 + 10.5806i) q^{5} -7.79840i q^{7} +0.0620886i q^{8} +(-42.3020 + 14.4431i) q^{10} -32.7050 q^{11} -65.9775i q^{13} +31.1784 q^{14} -64.1240 q^{16} +15.5853i q^{17} -51.6661 q^{19} +(-28.8440 - 84.4807i) q^{20} -130.756i q^{22} +49.6796i q^{23} +(-98.8994 + 76.4454i) q^{25} +263.782 q^{26} +62.2661i q^{28} -282.020 q^{29} -147.663 q^{31} -255.875i q^{32} -62.3111 q^{34} +(82.5119 - 28.1718i) q^{35} -122.498i q^{37} -206.564i q^{38} +(-0.656937 + 0.224296i) q^{40} +109.032 q^{41} +191.493i q^{43} +261.132 q^{44} -198.622 q^{46} -414.420i q^{47} +282.185 q^{49} +(-305.633 - 395.406i) q^{50} +526.796i q^{52} -236.381i q^{53} +(-118.147 - 346.039i) q^{55} +0.484192 q^{56} -1127.53i q^{58} +784.024 q^{59} -535.138 q^{61} -590.363i q^{62} +510.010 q^{64} +(698.084 - 238.345i) q^{65} -121.069i q^{67} -124.441i q^{68} +(112.633 + 329.888i) q^{70} -590.432 q^{71} +986.078i q^{73} +489.753 q^{74} +412.526 q^{76} +255.046i q^{77} +607.025 q^{79} +(-231.649 - 678.472i) q^{80} +435.915i q^{82} +1292.46i q^{83} +(-164.903 + 56.3023i) q^{85} -765.598 q^{86} -2.03061i q^{88} -1506.78 q^{89} -514.519 q^{91} -396.665i q^{92} +1656.87 q^{94} +(-186.645 - 546.659i) q^{95} +610.887i q^{97} +1128.19i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} - 15 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 - 15 * q^5	\( 8 q - 24 q^{4} - 15 q^{5} + 7 q^{10} + 42 q^{14} + 4 q^{16} + 118 q^{19} + 129 q^{20} + 17 q^{25} + 36 q^{26} - 318 q^{29} - 416 q^{31} + 638 q^{34} - 192 q^{35} - 265 q^{40} - 486 q^{41} - 852 q^{44} - 598 q^{46} + 350 q^{49} - 1143 q^{50} - 162 q^{55} - 1530 q^{56} + 1146 q^{59} - 398 q^{61} + 1640 q^{64} + 1833 q^{65} + 630 q^{70} + 1728 q^{71} - 1218 q^{74} - 3498 q^{76} + 2596 q^{79} - 1923 q^{80} - 233 q^{85} - 480 q^{86} - 1086 q^{89} - 2574 q^{91} + 1238 q^{94} - 1674 q^{95}+O(q^{100}) \) 8 * q - 24 * q^4 - 15 * q^5 + 7 * q^10 + 42 * q^14 + 4 * q^16 + 118 * q^19 + 129 * q^20 + 17 * q^25 + 36 * q^26 - 318 * q^29 - 416 * q^31 + 638 * q^34 - 192 * q^35 - 265 * q^40 - 486 * q^41 - 852 * q^44 - 598 * q^46 + 350 * q^49 - 1143 * q^50 - 162 * q^55 - 1530 * q^56 + 1146 * q^59 - 398 * q^61 + 1640 * q^64 + 1833 * q^65 + 630 * q^70 + 1728 * q^71 - 1218 * q^74 - 3498 * q^76 + 2596 * q^79 - 1923 * q^80 - 233 * q^85 - 480 * q^86 - 1086 * q^89 - 2574 * q^91 + 1238 * q^94 - 1674 * q^95

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