LMFDB - Embedded newform 405.4.b.c.244.2

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

$q$-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	$-0.750154\pi$
$2$		−	3.99806i		−	1.41353i	0.707450	−	0.706764i	$-0.249846\pi$
$3$		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	$-0.352057\pi$
$11$	32.7050			0.896448			0.448224	+	0.893921i	$0.352057\pi$
$12$		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	$-0.964538\pi$
$17$		−	15.5853i		−	0.222353i	0.993801	−	0.111176i	$-0.0354619\pi$
$18$		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	$-0.927699\pi$
$23$		−	49.6796i		−	0.450387i	0.974314	−	0.225194i	$-0.0723015\pi$
$24$		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	$-0.141413\pi$
$29$	282.020			1.80586			0.902929	+	0.429790i	$0.141413\pi$
$30$		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	$-0.566584\pi$
$41$	−109.032			−0.415314			−0.207657	+	0.978202i	$0.566584\pi$
$42$		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	$0.222343\pi$
$47$			414.420i			1.28616i	−0.765800	+	0.643078i	$0.777657\pi$
$48$		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	$0.0990962\pi$
$53$			236.381i			0.612631i	−0.951930	+	0.306315i	$0.900904\pi$
$54$		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	$-0.832688\pi$
$59$	−784.024			−1.73002			−0.865011	+	0.501753i	$0.832688\pi$
$60$		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	$-0.335732\pi$
$71$	590.432			0.986921			0.493460	+	0.869768i	$0.335732\pi$
$72$		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	$-0.673792\pi$
$83$		−	1292.46i		−	1.70923i	0.519260	−	0.854616i	$-0.326208\pi$
$84$		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	$-0.145528\pi$
$89$	1506.78			1.79459			0.897296	+	0.441428i	$0.145528\pi$
$90$		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	$-0.592923\pi$
$101$	−584.247			−0.575592			−0.287796	+	0.957692i	$0.592923\pi$
$102$		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	$0.0522835\pi$
$107$			361.964i			0.327032i	−0.986541	+	0.163516i	$0.947716\pi$
$108$		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	$-0.799259\pi$
$113$		−	1416.62i		−	1.17933i	0.807646	−	0.589667i	$-0.200741\pi$
$114$		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	$-0.566419\pi$
$131$	−621.192			−0.414304			−0.207152	+	0.978309i	$0.566419\pi$
$132$		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	$-0.864044\pi$
$137$		−	1328.54i		−	0.828502i	0.910163	−	0.414251i	$-0.135956\pi$
$138$		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	$-0.471812\pi$
$149$	321.699			0.176877			0.0884384	+	0.996082i	$0.471812\pi$
$150$		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	$-0.846671\pi$
$167$		−	1999.64i		−	0.926569i	0.886209	−	0.463285i	$-0.153329\pi$
$168$		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	$-0.764267\pi$
$173$		−	3070.57i		−	1.34943i	0.738079	−	0.674714i	$-0.235733\pi$
$174$		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	$-0.543783\pi$
$179$	−656.735			−0.274227			−0.137114	+	0.990555i	$0.543783\pi$
$180$		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	$-0.133874\pi$
$191$	4819.27			1.82571			0.912853	+	0.408288i	$0.133874\pi$
$192$		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	$-0.874016\pi$
$197$		−	2132.04i		−	0.771075i	0.922692	−	0.385537i	$-0.125984\pi$
$198$		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	$-0.980992\pi$
$227$		−	408.227i		−	0.119361i	0.998218	−	0.0596805i	$-0.0190082\pi$
$228$		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	$-0.870577\pi$
$233$		−	2813.15i		−	0.790968i	0.918473	−	0.395484i	$-0.129423\pi$
$234$		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	$-0.215734\pi$
$239$	5756.48			1.55797			0.778987	+	0.627040i	$0.215734\pi$
$240$		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	$-0.545010\pi$
$251$	−1120.86			−0.281866			−0.140933	+	0.990019i	$0.545010\pi$
$252$		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	$-0.961952\pi$
$257$		−	982.609i		−	0.238496i	0.992864	−	0.119248i	$-0.0380484\pi$
$258$		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	$0.131305\pi$
$263$			3419.85i			0.801813i	−0.916119	+	0.400907i	$0.868695\pi$
$264$		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	$-0.448344\pi$
$269$	1425.68			0.323142			0.161571	+	0.986861i	$0.448344\pi$
$270$		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	$-0.468311\pi$
$281$	936.334			0.198779			0.0993897	+	0.995049i	$0.468311\pi$
$282$		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	$-0.946823\pi$
$293$		−	1667.96i		−	0.332571i	0.986078	−	0.166285i	$-0.0531773\pi$
$294$		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	$-0.479751\pi$
$311$	697.317			0.127142			0.0635711	+	0.997977i	$0.479751\pi$
$312$		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	$-0.839018\pi$
$317$		−	5468.56i		−	0.968911i	0.874816	−	0.484455i	$-0.160982\pi$
$318$		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	$-0.786734\pi$
$347$		−	8027.94i		−	1.24197i	0.783824	−	0.620983i	$-0.213266\pi$
$348$		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	$0.103537\pi$
$353$			4238.90i			0.639133i	−0.947564	+	0.319567i	$0.896463\pi$
$354$		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	$-0.744554\pi$
$359$	−9453.60			−1.38981			−0.694905	+	0.719101i	$0.744554\pi$
$360$		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	$0.354730\pi$
$383$			13456.7i			1.79531i	−0.440701	+	0.897654i	$0.645270\pi$
$384$		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	$-0.322957\pi$
$389$	8101.33			1.05592			0.527961	+	0.849269i	$0.322957\pi$
$390$		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	$-0.445897\pi$
$401$	2716.60			0.338305			0.169153	+	0.985590i	$0.445897\pi$
$402$		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	$-0.633982\pi$
$419$	−7008.83			−0.817193			−0.408596	+	0.912715i	$0.633982\pi$
$420$		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	$-0.665188\pi$
$431$	−8875.68			−0.991941			−0.495970	+	0.868339i	$0.665188\pi$
$432$		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	$0.0877230\pi$
$443$			5074.43i			0.544229i	−0.962265	+	0.272115i	$0.912277\pi$
$444$		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	$-0.712558\pi$
$449$	−11783.0			−1.23848			−0.619238	+	0.785203i	$0.712558\pi$
$450$		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	$-0.367794\pi$
$461$	7987.75			0.806999			0.403499	+	0.914980i	$0.367794\pi$
$462$		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	$0.0961178\pi$
$467$			6002.60i			0.594790i	−0.954755	+	0.297395i	$0.903882\pi$
$468$		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	$-0.461701\pi$
$479$	2516.64			0.240059			0.120029	+	0.992770i	$0.461701\pi$
$480$		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	$-0.365688\pi$
$491$	8911.50			0.819084			0.409542	+	0.912291i	$0.365688\pi$
$492$		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	$0.192833\pi$
$503$			12847.4i			1.13884i	−0.822046	+	0.569422i	$0.807167\pi$
$504$		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	$-0.444887\pi$
$509$	3956.77			0.344560			0.172280	+	0.985048i	$0.444887\pi$
$510$		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	$-0.413389\pi$
$521$	6392.00			0.537502			0.268751	+	0.963210i	$0.413389\pi$
$522$		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	$-0.946155\pi$
$557$		−	4426.25i		−	0.336708i	0.985727	−	0.168354i	$-0.0538451\pi$
$558$		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	$0.0378957\pi$
$563$			3173.26i			0.237543i	−0.992922	+	0.118772i	$0.962104\pi$
$564$		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	$-0.543117\pi$
$569$	−3665.79			−0.270084			−0.135042	+	0.990840i	$0.543117\pi$
$570$		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	$-0.779082\pi$
$587$		−	18193.9i		−	1.27929i	0.768671	−	0.639645i	$-0.220918\pi$
$588$		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	$-0.995055\pi$
$593$		−	448.648i		−	0.0310688i	0.999879	−	0.0155344i	$-0.00494495\pi$
$594$		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	$-0.403108\pi$
$599$	8787.84			0.599435			0.299717	+	0.954028i	$0.403108\pi$
$600$		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	$-0.732360\pi$
$617$		−	22841.4i		−	1.49038i	0.666854	−	0.745188i	$-0.267640\pi$
$618$		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	$-0.528609\pi$
$641$	−2913.25			−0.179511			−0.0897555	+	0.995964i	$0.528609\pi$
$642$		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	$-0.917496\pi$
$647$		−	8436.06i		−	0.512605i	0.966597	−	0.256303i	$-0.0825044\pi$
$648$		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	$-0.828245\pi$
$653$		−	17146.6i		−	1.02756i	0.857922	−	0.513780i	$-0.171755\pi$
$654$		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	$-0.516703\pi$
$659$	−1774.60			−0.104900			−0.0524498	+	0.998624i	$0.516703\pi$
$660$		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	$-0.959577\pi$
$677$		−	4461.98i		−	0.253306i	0.991947	−	0.126653i	$-0.0404234\pi$
$678$		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	$0.386926\pi$
$683$			33470.5i			1.87513i	−0.347808	+	0.937566i	$0.613074\pi$
$684$		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	$-0.467682\pi$
$701$	3762.36			0.202714			0.101357	+	0.994850i	$0.467682\pi$
$702$		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	$-0.530995\pi$
$719$	−3748.68			−0.194440			−0.0972200	+	0.995263i	$0.530995\pi$
$720$		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	$0.257209\pi$
$743$			29282.9i			1.44588i	−0.690913	+	0.722938i	$0.742791\pi$
$744$		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	$-0.835946\pi$
$761$	−36532.3			−1.74020			−0.870101	+	0.492874i	$0.835946\pi$
$762$		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	$-0.773376\pi$
$773$		−	28081.8i		−	1.30664i	0.757083	−	0.653319i	$-0.226624\pi$
$774$		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	$-0.973314\pi$
$797$		−	3768.31i		−	0.167478i	0.996488	−	0.0837392i	$-0.0266863\pi$
$798$		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	$-0.676949\pi$
$809$	−24285.5			−1.05542			−0.527708	+	0.849426i	$0.676949\pi$
$810$		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	$-0.562375\pi$
$821$	−9160.55			−0.389410			−0.194705	+	0.980862i	$0.562375\pi$
$822$		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	$0.175206\pi$
$827$			24878.9i			1.04610i	−0.852302	+	0.523050i	$0.824794\pi$
$828$		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	$-0.539668\pi$
$839$	−6041.34			−0.248594			−0.124297	+	0.992245i	$0.539668\pi$
$840$		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	$-0.754406\pi$
$857$		−	34985.7i		−	1.39450i	0.716826	−	0.697252i	$-0.245594\pi$
$858$		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	$-0.947608\pi$
$863$		−	8308.05i		−	0.327705i	0.986485	−	0.163852i	$-0.0523921\pi$
$864$		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	$-0.326068\pi$
$881$	27176.4			1.03927			0.519634	+	0.854389i	$0.326068\pi$
$882$		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	$0.0588276\pi$
$887$			9708.93i			0.367524i	−0.982971	+	0.183762i	$0.941172\pi$
$888$		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	$-0.760638\pi$
$911$	−40163.5			−1.46068			−0.730338	+	0.683086i	$0.760638\pi$
$912$		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	$-0.520757\pi$
$929$	−3690.27			−0.130327			−0.0651636	+	0.997875i	$0.520757\pi$
$930$		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	$-0.329942\pi$
$941$	29396.9			1.01840			0.509198	+	0.860650i	$0.329942\pi$
$942$		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	$-0.991246\pi$
$947$		−	1602.80i		−	0.0549991i	0.999622	−	0.0274995i	$-0.00875448\pi$
$948$		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	$-0.918685\pi$
$953$		−	14868.1i		−	0.505377i	0.967548	−	0.252689i	$-0.0813148\pi$
$954$		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	$-0.181364\pi$
$971$	50954.6			1.68405			0.842024	+	0.539440i	$0.181364\pi$
$972$		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	$0.0986456\pi$
$977$			18626.3i			0.609935i	−0.952363	+	0.304968i	$0.901354\pi$
$978$		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	$-0.940227\pi$
$983$		−	11506.9i		−	0.373361i	0.982421	−	0.186681i	$-0.0597730\pi$
$984$		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.c.244.2	yes	8
3.2	odd	2		405.4.b.b.244.7	yes	8
5.2	odd	4		2025.4.a.bd.1.7		8
5.3	odd	4		2025.4.a.bd.1.2		8
5.4	even	2	inner	405.4.b.c.244.7	yes	8
15.2	even	4		2025.4.a.bc.1.2		8
15.8	even	4		2025.4.a.bc.1.7		8
15.14	odd	2		405.4.b.b.244.2	✓	8

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