LMFDB - Embedded newform 825.3.b.a.76.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,3,Mod(76,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.76");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	825.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:	$22.4796218097$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.39744.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 12x^{2} + 4x + 22 $ x^4 - 2x^3 + 12x^2 + 4*x + 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.4
Root			$-0.366025 + 1.29224i$ of defining polynomial
Character	$\chi$	$=$	825.76
Dual form			825.3.b.a.76.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.53045i q^{2} +1.73205 q^{3} -8.46410 q^{4} +6.11492i q^{6} +2.58447i q^{7} -15.7603i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+3.53045i q^{2} +1.73205 q^{3} -8.46410 q^{4} +6.11492i q^{6} +2.58447i q^{7} -15.7603i q^{8} +3.00000 q^{9} +(6.73205 - 8.69940i) q^{11} -14.6603 q^{12} -23.7672i q^{13} -9.12436 q^{14} +21.7846 q^{16} -12.2298i q^{17} +10.5914i q^{18} +3.27698i q^{19} +4.47644i q^{21} +(30.7128 + 23.7672i) q^{22} +14.3397 q^{23} -27.2976i q^{24} +83.9090 q^{26} +5.19615 q^{27} -21.8752i q^{28} -38.5815i q^{29} -11.1769 q^{31} +13.8683i q^{32} +(11.6603 - 15.0678i) q^{33} +43.1769 q^{34} -25.3923 q^{36} +12.5359 q^{37} -11.5692 q^{38} -41.1660i q^{39} -1.38501i q^{41} -15.8038 q^{42} -23.9527i q^{43} +(-56.9808 + 73.6326i) q^{44} +50.6258i q^{46} -19.8038 q^{47} +37.7321 q^{48} +42.3205 q^{49} -21.1827i q^{51} +201.168i q^{52} +12.0526 q^{53} +18.3448i q^{54} +40.7321 q^{56} +5.67589i q^{57} +136.210 q^{58} -62.7461 q^{59} -21.3683i q^{61} -39.4596i q^{62} +7.75341i q^{63} +38.1769 q^{64} +(53.1962 + 41.1660i) q^{66} +34.0000 q^{67} +103.515i q^{68} +24.8372 q^{69} -69.2679 q^{71} -47.2809i q^{72} +39.9665i q^{73} +44.2574i q^{74} -27.7367i q^{76} +(22.4833 + 17.3988i) q^{77} +145.335 q^{78} +97.6532i q^{79} +9.00000 q^{81} +4.88973 q^{82} +71.9941i q^{83} -37.8890i q^{84} +84.5641 q^{86} -66.8251i q^{87} +(-137.105 - 106.099i) q^{88} +107.177 q^{89} +61.4256 q^{91} -121.373 q^{92} -19.3590 q^{93} -69.9166i q^{94} +24.0207i q^{96} +166.746 q^{97} +149.411i q^{98} +(20.1962 - 26.0982i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{4} + 12 q^{9}+O(q^{10}) $ 4 * q - 20 * q^4 + 12 * q^9	$ 4 q - 20 q^{4} + 12 q^{9} + 20 q^{11} - 24 q^{12} + 12 q^{14} + 4 q^{16} + 12 q^{22} + 92 q^{23} + 204 q^{26} + 80 q^{31} + 12 q^{33} + 48 q^{34} - 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42} - 124 q^{44} - 100 q^{47} + 144 q^{48} + 100 q^{49} - 28 q^{53} + 156 q^{56} + 240 q^{58} + 40 q^{59} + 28 q^{64} + 192 q^{66} + 136 q^{67} - 60 q^{69} - 284 q^{71} + 180 q^{77} + 228 q^{78} + 36 q^{81} - 216 q^{82} - 216 q^{86} - 396 q^{88} + 304 q^{89} + 24 q^{91} - 340 q^{92} - 216 q^{93} + 376 q^{97} + 60 q^{99}+O(q^{100}) $ 4 * q - 20 * q^4 + 12 * q^9 + 20 * q^11 - 24 * q^12 + 12 * q^14 + 4 * q^16 + 12 * q^22 + 92 * q^23 + 204 * q^26 + 80 * q^31 + 12 * q^33 + 48 * q^34 - 60 * q^36 + 64 * q^37 + 120 * q^38 - 84 * q^42 - 124 * q^44 - 100 * q^47 + 144 * q^48 + 100 * q^49 - 28 * q^53 + 156 * q^56 + 240 * q^58 + 40 * q^59 + 28 * q^64 + 192 * q^66 + 136 * q^67 - 60 * q^69 - 284 * q^71 + 180 * q^77 + 228 * q^78 + 36 * q^81 - 216 * q^82 - 216 * q^86 - 396 * q^88 + 304 * q^89 + 24 * q^91 - 340 * q^92 - 216 * q^93 + 376 * q^97 + 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			3.53045i			1.76523i	0.470100	+	0.882613i	$0.344218\pi$
$2$			3.53045i			1.76523i	−0.470100	+	0.882613i	$0.655782\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−8.46410			−2.11603
$5$		0			0
$5$		0			0
$6$			6.11492i			1.01915i
$7$			2.58447i			0.369210i	0.982813	+	0.184605i	$0.0591006\pi$
$7$			2.58447i			0.369210i	−0.982813	+	0.184605i	$0.940899\pi$
$8$		−	15.7603i		−	1.97004i
$9$	3.00000			0.333333
$10$		0			0
$11$	6.73205	−	8.69940i	0.612005	−	0.790854i
$11$	6.73205	−	8.69940i	0.612005	−	0.790854i
$12$	−14.6603			−1.22169
$13$		−	23.7672i		−	1.82825i	−0.405437	−	0.914123i	$-0.632881\pi$
$13$		−	23.7672i		−	1.82825i	0.405437	−	0.914123i	$-0.367119\pi$
$14$	−9.12436			−0.651740
$15$		0			0
$16$	21.7846			1.36154
$17$		−	12.2298i		−	0.719403i	−0.933067	−	0.359701i	$-0.882879\pi$
$17$		−	12.2298i		−	0.719403i	0.933067	−	0.359701i	$-0.117121\pi$
$18$			10.5914i			0.588409i
$19$			3.27698i			0.172473i	0.996275	+	0.0862363i	$0.0274840\pi$
$19$			3.27698i			0.172473i	−0.996275	+	0.0862363i	$0.972516\pi$
$20$		0			0
$21$			4.47644i			0.213164i
$22$	30.7128	+	23.7672i	1.39604	+	1.08033i
$23$	14.3397			0.623467			0.311734	−	0.950170i	$-0.399090\pi$
$23$	14.3397			0.623467			0.311734	+	0.950170i	$0.399090\pi$
$24$		−	27.2976i		−	1.13740i
$25$		0			0
$26$	83.9090			3.22727
$27$	5.19615			0.192450
$28$		−	21.8752i		−	0.781258i
$29$		−	38.5815i		−	1.33040i	−0.746667	−	0.665198i	$-0.768347\pi$
$29$		−	38.5815i		−	1.33040i	0.746667	−	0.665198i	$-0.231653\pi$
$30$		0			0
$31$	−11.1769			−0.360546			−0.180273	−	0.983617i	$-0.557698\pi$
$31$	−11.1769			−0.360546			−0.180273	+	0.983617i	$0.557698\pi$
$32$			13.8683i			0.433386i
$33$	11.6603	−	15.0678i	0.353341	−	0.456600i
$34$	43.1769			1.26991
$35$		0			0
$36$	−25.3923			−0.705342
$37$	12.5359			0.338808			0.169404	−	0.985547i	$-0.445816\pi$
$37$	12.5359			0.338808			0.169404	+	0.985547i	$0.445816\pi$
$38$	−11.5692			−0.304453
$39$		−	41.1660i		−	1.05554i
$40$		0			0
$41$		−	1.38501i		−	0.0337808i	−0.999857	−	0.0168904i	$-0.994623\pi$
$41$		−	1.38501i		−	0.0337808i	0.999857	−	0.0168904i	$-0.00537664\pi$
$42$	−15.8038			−0.376282
$43$		−	23.9527i		−	0.557041i	−0.960430	−	0.278520i	$-0.910156\pi$
$43$		−	23.9527i		−	0.557041i	0.960430	−	0.278520i	$-0.0898440\pi$
$44$	−56.9808	+	73.6326i	−1.29502	+	1.67347i
$45$		0			0
$46$			50.6258i			1.10056i
$47$	−19.8038			−0.421358			−0.210679	−	0.977555i	$-0.567568\pi$
$47$	−19.8038			−0.421358			−0.210679	+	0.977555i	$0.567568\pi$
$48$	37.7321			0.786084
$49$	42.3205			0.863684
$50$		0			0
$51$		−	21.1827i		−	0.415347i
$52$			201.168i			3.86861i
$53$	12.0526			0.227407			0.113703	−	0.993515i	$-0.463729\pi$
$53$	12.0526			0.227407			0.113703	+	0.993515i	$0.463729\pi$
$54$			18.3448i			0.339718i
$55$		0			0
$56$	40.7321			0.727358
$57$			5.67589i			0.0995771i
$58$	136.210			2.34845
$59$	−62.7461			−1.06349			−0.531747	−	0.846903i	$-0.678464\pi$
$59$	−62.7461			−1.06349			−0.531747	+	0.846903i	$0.678464\pi$
$60$		0			0
$61$		−	21.3683i		−	0.350300i	−0.984542	−	0.175150i	$-0.943959\pi$
$61$		−	21.3683i		−	0.350300i	0.984542	−	0.175150i	$-0.0560410\pi$
$62$		−	39.4596i		−	0.636445i
$63$			7.75341i			0.123070i
$64$	38.1769			0.596514
$65$		0			0
$66$	53.1962	+	41.1660i	0.806002	+	0.623727i
$67$	34.0000			0.507463			0.253731	−	0.967275i	$-0.418342\pi$
$67$	34.0000			0.507463			0.253731	+	0.967275i	$0.418342\pi$
$68$			103.515i			1.52227i
$69$	24.8372			0.359959
$70$		0			0
$71$	−69.2679			−0.975605			−0.487802	−	0.872954i	$-0.662201\pi$
$71$	−69.2679			−0.975605			−0.487802	+	0.872954i	$0.662201\pi$
$72$		−	47.2809i		−	0.656679i
$73$			39.9665i			0.547487i	0.961803	+	0.273743i	$0.0882619\pi$
$73$			39.9665i			0.547487i	−0.961803	+	0.273743i	$0.911738\pi$
$74$			44.2574i			0.598073i
$75$		0			0
$76$		−	27.7367i		−	0.364956i
$77$	22.4833	+	17.3988i	0.291991	+	0.225958i
$78$	145.335			1.86326
$79$			97.6532i			1.23612i	0.786132	+	0.618058i	$0.212081\pi$
$79$			97.6532i			1.23612i	−0.786132	+	0.618058i	$0.787919\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	4.88973			0.0596308
$83$			71.9941i			0.867399i	0.901058	+	0.433699i	$0.142792\pi$
$83$			71.9941i			0.867399i	−0.901058	+	0.433699i	$0.857208\pi$
$84$		−	37.8890i		−	0.451060i
$85$		0			0
$86$	84.5641			0.983303
$87$		−	66.8251i		−	0.768105i
$88$	−137.105	−	106.099i	−1.55801	−	1.20567i
$89$	107.177			1.20423			0.602117	−	0.798407i	$-0.294324\pi$
$89$	107.177			1.20423			0.602117	+	0.798407i	$0.294324\pi$
$90$		0			0
$91$	61.4256			0.675007
$92$	−121.373			−1.31927
$93$	−19.3590			−0.208161
$94$		−	69.9166i		−	0.743793i
$95$		0			0
$96$			24.0207i			0.250215i
$97$	166.746			1.71903			0.859516	−	0.511109i	$-0.170765\pi$
$97$	166.746			1.71903			0.859516	+	0.511109i	$0.170765\pi$
$98$			149.411i			1.52460i
$99$	20.1962	−	26.0982i	0.204002	−	0.263618i
$100$		0			0
$101$			92.6699i			0.917523i	0.888559	+	0.458762i	$0.151707\pi$
$101$			92.6699i			0.917523i	−0.888559	+	0.458762i	$0.848293\pi$
$102$	74.7846			0.733182
$103$	−140.746			−1.36647			−0.683234	−	0.730200i	$-0.739427\pi$
$103$	−140.746			−1.36647			−0.683234	+	0.730200i	$0.739427\pi$
$104$	−374.578			−3.60171
$105$		0			0
$106$			42.5510i			0.401425i
$107$			74.8999i			0.700000i	0.936750	+	0.350000i	$0.113818\pi$
$107$			74.8999i			0.700000i	−0.936750	+	0.350000i	$0.886182\pi$
$108$	−43.9808			−0.407229
$109$		−	153.634i		−	1.40948i	−0.709465	−	0.704741i	$-0.751063\pi$
$109$		−	153.634i		−	1.40948i	0.709465	−	0.704741i	$-0.248937\pi$
$110$		0			0
$111$	21.7128			0.195611
$112$			56.3017i			0.502694i
$113$	−166.315			−1.47182			−0.735909	−	0.677081i	$-0.763245\pi$
$113$	−166.315			−1.47182			−0.735909	+	0.677081i	$0.763245\pi$
$114$	−20.0385			−0.175776
$115$		0			0
$116$			326.558i			2.81515i
$117$		−	71.3016i		−	0.609415i
$118$		−	221.522i		−	1.87731i
$119$	31.6077			0.265611
$120$		0			0
$121$	−30.3590	−	117.130i	−0.250901	−	0.968013i
$122$	75.4397			0.618358
$123$		−	2.39891i		−	0.0195034i
$124$	94.6025			0.762924
$125$		0			0
$126$	−27.3731			−0.217247
$127$		−	190.323i		−	1.49861i	−0.662227	−	0.749304i	$-0.730388\pi$
$127$		−	190.323i		−	1.49861i	0.662227	−	0.749304i	$-0.269612\pi$
$128$			190.255i			1.48637i
$129$		−	41.4874i		−	0.321608i
$130$		0			0
$131$		−	99.2238i		−	0.757434i	−0.925513	−	0.378717i	$-0.876365\pi$
$131$		−	99.2238i		−	0.757434i	0.925513	−	0.378717i	$-0.123635\pi$
$132$	−98.6936	+	127.535i	−0.747679	+	0.966177i
$133$	−8.46926			−0.0636786
$134$			120.035i			0.895787i
$135$		0			0
$136$	−192.746			−1.41725
$137$	29.2154			0.213251			0.106626	−	0.994299i	$-0.465995\pi$
$137$	29.2154			0.213251			0.106626	+	0.994299i	$0.465995\pi$
$138$			87.6865i			0.635409i
$139$		−	171.354i		−	1.23276i	−0.787449	−	0.616380i	$-0.788598\pi$
$139$		−	171.354i		−	1.23276i	0.787449	−	0.616380i	$-0.211402\pi$
$140$		0			0
$141$	−34.3013			−0.243271
$142$		−	244.547i		−	1.72216i
$143$	−206.760	−	160.002i	−1.44588	−	1.11889i
$144$	65.3538			0.453846
$145$		0			0
$146$	−141.100			−0.966438
$147$	73.3013			0.498648
$148$	−106.105			−0.716926
$149$		−	164.293i		−	1.10264i	−0.834295	−	0.551318i	$-0.814125\pi$
$149$		−	164.293i		−	1.10264i	0.834295	−	0.551318i	$-0.185875\pi$
$150$		0			0
$151$			229.547i			1.52018i	0.649817	+	0.760091i	$0.274846\pi$
$151$			229.547i			1.52018i	−0.649817	+	0.760091i	$0.725154\pi$
$152$	51.6462			0.339777
$153$		−	36.6895i		−	0.239801i
$154$	−61.4256	+	79.3764i	−0.398868	+	0.515431i
$155$		0			0
$156$			348.433i			2.23355i
$157$	106.862			0.680647			0.340323	−	0.940308i	$-0.389463\pi$
$157$	106.862			0.680647			0.340323	+	0.940308i	$0.389463\pi$
$158$	−344.760			−2.18203
$159$	20.8756			0.131293
$160$		0			0
$161$			37.0607i			0.230190i
$162$			31.7741i			0.196136i
$163$	118.067			0.724335			0.362168	−	0.932113i	$-0.382037\pi$
$163$	118.067			0.724335			0.362168	+	0.932113i	$0.382037\pi$
$164$			11.7229i			0.0714811i
$165$		0			0
$166$	−254.172			−1.53116
$167$			1.38501i			0.00829350i	0.999991	+	0.00414675i	$0.00131995\pi$
$167$			1.38501i			0.00829350i	−0.999991	+	0.00414675i	$0.998680\pi$
$168$	70.5500			0.419940
$169$	−395.879			−2.34248
$170$		0			0
$171$			9.83093i			0.0574908i
$172$			202.738i			1.17871i
$173$			67.8390i			0.392133i	0.980591	+	0.196067i	$0.0628169\pi$
$173$			67.8390i			0.392133i	−0.980591	+	0.196067i	$0.937183\pi$
$174$	235.923			1.35588
$175$		0			0
$176$	146.655	−	189.513i	0.833268	−	1.07678i
$177$	−108.679			−0.614008
$178$			378.383i			2.12575i
$179$	−136.603			−0.763143			−0.381571	−	0.924339i	$-0.624617\pi$
$179$	−136.603			−0.763143			−0.381571	+	0.924339i	$0.624617\pi$
$180$		0			0
$181$	172.172			0.951225			0.475613	−	0.879655i	$-0.342226\pi$
$181$	172.172			0.951225			0.475613	+	0.879655i	$0.342226\pi$
$182$			216.860i			1.19154i
$183$		−	37.0109i		−	0.202246i
$184$		−	225.999i		−	1.22825i
$185$		0			0
$186$		−	68.3460i		−	0.367452i
$187$	−106.392	−	82.3320i	−0.568943	−	0.440278i
$188$	167.622			0.891605
$189$			13.4293i			0.0710545i
$190$		0			0
$191$	137.765			0.721285			0.360642	−	0.932704i	$-0.382558\pi$
$191$	137.765			0.721285			0.360642	+	0.932704i	$0.382558\pi$
$192$	66.1244			0.344398
$193$		−	106.421i		−	0.551402i	−0.961243	−	0.275701i	$-0.911090\pi$
$193$		−	106.421i		−	0.551402i	0.961243	−	0.275701i	$-0.0889099\pi$
$194$			588.689i			3.03448i
$195$		0			0
$196$	−358.205			−1.82758
$197$			33.4126i			0.169607i	0.996398	+	0.0848035i	$0.0270262\pi$
$197$			33.4126i			0.169607i	−0.996398	+	0.0848035i	$0.972974\pi$
$198$	92.1384	+	71.3016i	0.465346	+	0.360109i
$199$	334.056			1.67868			0.839338	−	0.543611i	$-0.182943\pi$
$199$	334.056			1.67868			0.839338	+	0.543611i	$0.182943\pi$
$200$		0			0
$201$	58.8897			0.292984
$202$	−327.167			−1.61964
$203$	99.7128			0.491196
$204$			179.293i			0.878886i
$205$		0			0
$206$		−	496.898i		−	2.41212i
$207$	43.0192			0.207822
$208$		−	517.759i		−	2.48923i
$209$	28.5077	+	22.0608i	0.136401	+	0.105554i
$210$		0			0
$211$		−	31.1495i		−	0.147628i	−0.997272	−	0.0738140i	$-0.976483\pi$
$211$		−	31.1495i		−	0.147628i	0.997272	−	0.0738140i	$-0.0235171\pi$
$212$	−102.014			−0.481199
$213$	−119.976			−0.563266
$214$	−264.431			−1.23566
$215$		0			0
$216$		−	81.8929i		−	0.379134i
$217$		−	28.8864i		−	0.133117i
$218$	542.396			2.48806
$219$			69.2241i			0.316092i
$220$		0			0
$221$	−290.669			−1.31525
$222$			76.6561i			0.345298i
$223$	−137.713			−0.617546			−0.308773	−	0.951136i	$-0.599918\pi$
$223$	−137.713			−0.617546			−0.308773	+	0.951136i	$0.599918\pi$
$224$	−35.8423			−0.160010
$225$		0			0
$226$		−	587.169i		−	2.59809i
$227$		−	329.835i		−	1.45302i	−0.687158	−	0.726508i	$-0.741142\pi$
$227$		−	329.835i		−	1.45302i	0.687158	−	0.726508i	$-0.258858\pi$
$228$		−	48.0413i		−	0.210708i
$229$	299.282			1.30691			0.653454	−	0.756966i	$-0.273319\pi$
$229$	299.282			1.30691			0.653454	+	0.756966i	$0.273319\pi$
$230$		0			0
$231$	38.9423	+	30.1356i	0.168581	+	0.130457i
$232$	−608.056			−2.62093
$233$			146.115i			0.627105i	0.949571	+	0.313552i	$0.101519\pi$
$233$			146.115i			0.627105i	−0.949571	+	0.313552i	$0.898481\pi$
$234$	251.727			1.07576
$235$		0			0
$236$	531.090			2.25038
$237$			169.140i			0.713672i
$238$			111.589i			0.468863i
$239$		−	63.4123i		−	0.265324i	−0.991161	−	0.132662i	$-0.957648\pi$
$239$		−	63.4123i		−	0.265324i	0.991161	−	0.132662i	$-0.0423524\pi$
$240$		0			0
$241$		−	351.018i		−	1.45650i	−0.685309	−	0.728252i	$-0.740333\pi$
$241$		−	351.018i		−	1.45650i	0.685309	−	0.728252i	$-0.259667\pi$
$242$	413.520	−	107.181i	1.70876	−	0.442897i
$243$	15.5885			0.0641500
$244$			180.863i			0.741243i
$245$		0			0
$246$	8.46926			0.0344279
$247$	77.8846			0.315322
$248$			176.152i			0.710289i
$249$			124.697i			0.500793i
$250$		0			0
$251$	65.7231			0.261845			0.130923	−	0.991393i	$-0.458206\pi$
$251$	65.7231			0.261845			0.130923	+	0.991393i	$0.458206\pi$
$252$		−	65.6257i		−	0.260419i
$253$	96.5359	−	124.747i	0.381565	−	0.493072i
$254$	671.927			2.64538
$255$		0			0
$256$	−518.979			−2.02726
$257$	206.182			0.802265			0.401132	−	0.916020i	$-0.368617\pi$
$257$	206.182			0.802265			0.401132	+	0.916020i	$0.368617\pi$
$258$	146.469			0.567710
$259$			32.3987i			0.125091i
$260$		0			0
$261$		−	115.745i		−	0.443466i
$262$	350.305			1.33704
$263$			18.2769i			0.0694937i	0.999396	+	0.0347469i	$0.0110625\pi$
$263$			18.2769i			0.0694937i	−0.999396	+	0.0347469i	$0.988937\pi$
$264$	−237.473	−	183.769i	−0.899519	−	0.696095i
$265$		0			0
$266$		−	29.9003i		−	0.112407i
$267$	185.636			0.695265
$268$	−287.779			−1.07380
$269$	−172.130			−0.639887			−0.319943	−	0.947437i	$-0.603664\pi$
$269$	−172.130			−0.639887			−0.319943	+	0.947437i	$0.603664\pi$
$270$		0			0
$271$			425.967i			1.57183i	0.618332	+	0.785917i	$0.287809\pi$
$271$			425.967i			1.57183i	−0.618332	+	0.785917i	$0.712191\pi$
$272$		−	266.422i		−	0.979494i
$273$	106.392			0.389715
$274$			103.144i			0.376436i
$275$		0			0
$276$	−210.224			−0.761682
$277$		−	261.439i		−	0.943824i	−0.881646	−	0.471912i	$-0.843564\pi$
$277$		−	261.439i		−	0.943824i	0.881646	−	0.471912i	$-0.156436\pi$
$278$	604.956			2.17610
$279$	−33.5307			−0.120182
$280$		0			0
$281$			329.192i			1.17150i	0.810491	+	0.585751i	$0.199200\pi$
$281$			329.192i			1.17150i	−0.810491	+	0.585751i	$0.800800\pi$
$282$		−	121.099i		−	0.429429i
$283$		−	231.353i		−	0.817503i	−0.912646	−	0.408751i	$-0.865964\pi$
$283$		−	231.353i		−	0.817503i	0.912646	−	0.408751i	$-0.134036\pi$
$284$	586.291			2.06440
$285$		0			0
$286$	564.879	−	729.957i	1.97510	−	2.55230i
$287$	3.57953			0.0124722
$288$			41.6050i			0.144462i
$289$	139.431			0.482459
$290$		0			0
$291$	288.813			0.992484
$292$		−	338.281i		−	1.15850i
$293$		−	199.733i		−	0.681683i	−0.940121	−	0.340842i	$-0.889288\pi$
$293$		−	199.733i		−	0.681683i	0.940121	−	0.340842i	$-0.110712\pi$
$294$			258.787i			0.880227i
$295$		0			0
$296$		−	197.570i		−	0.667465i
$297$	34.9808	−	45.2034i	0.117780	−	0.152200i
$298$	580.028			1.94640
$299$		−	340.815i		−	1.13985i
$300$		0			0
$301$	61.9052			0.205665
$302$	−810.406			−2.68346
$303$			160.509i			0.529732i
$304$			71.3877i			0.234828i
$305$		0			0
$306$	129.531			0.423303
$307$			426.289i			1.38856i	0.719704	+	0.694281i	$0.244278\pi$
$307$			426.289i			1.38856i	−0.719704	+	0.694281i	$0.755722\pi$
$308$	−190.301	−	147.265i	−0.617861	−	0.478134i
$309$	−243.779			−0.788930
$310$		0			0
$311$	−301.611			−0.969812			−0.484906	−	0.874566i	$-0.661146\pi$
$311$	−301.611			−0.969812			−0.484906	+	0.874566i	$0.661146\pi$
$312$	−648.788			−2.07945
$313$	−260.697			−0.832899			−0.416449	−	0.909159i	$-0.636726\pi$
$313$	−260.697			−0.832899			−0.416449	+	0.909159i	$0.636726\pi$
$314$			377.270i			1.20150i
$315$		0			0
$316$		−	826.547i		−	2.61565i
$317$	−451.191			−1.42332			−0.711658	−	0.702526i	$-0.752055\pi$
$317$	−451.191			−1.42332			−0.711658	+	0.702526i	$0.752055\pi$
$318$			73.7005i			0.231763i
$319$	−335.636	−	259.733i	−1.05215	−	0.814209i
$320$		0			0
$321$			129.731i			0.404145i
$322$	−130.841			−0.406338
$323$	40.0770			0.124077
$324$	−76.1769			−0.235114
$325$		0			0
$326$			416.829i			1.27862i
$327$		−	266.101i		−	0.813765i
$328$	−21.8282			−0.0665495
$329$		−	51.1825i		−	0.155570i
$330$		0			0
$331$	229.569			0.693563			0.346781	−	0.937946i	$-0.387275\pi$
$331$	229.569			0.693563			0.346781	+	0.937946i	$0.387275\pi$
$332$		−	609.365i		−	1.83544i
$333$	37.6077			0.112936
$334$	−4.88973			−0.0146399
$335$		0			0
$336$			97.5174i			0.290230i
$337$			178.143i			0.528614i	0.964439	+	0.264307i	$0.0851432\pi$
$337$			178.143i			0.528614i	−0.964439	+	0.264307i	$0.914857\pi$
$338$		−	1397.63i		−	4.13501i
$339$	−288.067			−0.849754
$340$		0			0
$341$	−75.2436	+	97.2324i	−0.220656	+	0.285139i
$342$	−34.7077			−0.101484
$343$			236.015i			0.688091i
$344$	−377.503			−1.09739
$345$		0			0
$346$	−239.503			−0.692204
$347$			574.839i			1.65660i	0.560287	+	0.828299i	$0.310691\pi$
$347$			574.839i			1.65660i	−0.560287	+	0.828299i	$0.689309\pi$
$348$			565.615i			1.62533i
$349$			182.619i			0.523265i	0.965168	+	0.261632i	$0.0842608\pi$
$349$			182.619i			0.523265i	−0.965168	+	0.261632i	$0.915739\pi$
$350$		0			0
$351$		−	123.498i		−	0.351846i
$352$	120.646	+	93.3624i	0.342745	+	0.265234i
$353$	195.990			0.555212			0.277606	−	0.960695i	$-0.410459\pi$
$353$	195.990			0.555212			0.277606	+	0.960695i	$0.410459\pi$
$354$		−	383.688i		−	1.08386i
$355$		0			0
$356$	−907.156			−2.54819
$357$	54.7461			0.153351
$358$		−	482.269i		−	1.34712i
$359$		−	707.105i		−	1.96965i	−0.173548	−	0.984825i	$-0.555523\pi$
$359$		−	707.105i		−	1.96965i	0.173548	−	0.984825i	$-0.444477\pi$
$360$		0			0
$361$	350.261			0.970253
$362$			607.844i			1.67913i
$363$	−52.5833	−	202.874i	−0.144858	−	0.558882i
$364$	−519.913			−1.42833
$365$		0			0
$366$	130.665			0.357009
$367$	−359.415			−0.979333			−0.489667	−	0.871910i	$-0.662882\pi$
$367$	−359.415			−0.979333			−0.489667	+	0.871910i	$0.662882\pi$
$368$	312.386			0.848874
$369$		−	4.15504i		−	0.0112603i
$370$		0			0
$371$			31.1495i			0.0839609i
$372$	163.856			0.440474
$373$		−	363.804i		−	0.975346i	−0.873026	−	0.487673i	$-0.837846\pi$
$373$		−	363.804i		−	0.975346i	0.873026	−	0.487673i	$-0.162154\pi$
$374$	290.669	−	375.613i	0.777190	−	1.00431i
$375$		0			0
$376$			312.115i			0.830092i
$377$	−916.974			−2.43229
$378$	−47.4115			−0.125427
$379$	−266.046			−0.701968			−0.350984	−	0.936381i	$-0.614153\pi$
$379$	−266.046			−0.701968			−0.350984	+	0.936381i	$0.614153\pi$
$380$		0			0
$381$		−	329.649i		−	0.865221i
$382$			486.374i			1.27323i
$383$	−331.258			−0.864902			−0.432451	−	0.901657i	$-0.642351\pi$
$383$	−331.258			−0.864902			−0.432451	+	0.901657i	$0.642351\pi$
$384$			329.532i			0.858155i
$385$		0			0
$386$	375.713			0.973349
$387$		−	71.8582i		−	0.185680i
$388$	−1411.36			−3.63752
$389$	421.027			1.08233			0.541166	−	0.840916i	$-0.317983\pi$
$389$	421.027			1.08233			0.541166	+	0.840916i	$0.317983\pi$
$390$		0			0
$391$		−	175.373i		−	0.448524i
$392$		−	666.984i		−	1.70149i
$393$		−	171.861i		−	0.437305i
$394$	−117.962			−0.299395
$395$		0			0
$396$	−170.942	+	220.898i	−0.431672	+	0.557823i
$397$	421.797			1.06246			0.531231	−	0.847227i	$-0.321730\pi$
$397$	421.797			1.06246			0.531231	+	0.847227i	$0.321730\pi$
$398$			1179.37i			2.96324i
$399$	−14.6692			−0.0367649
$400$		0			0
$401$	−353.531			−0.881623			−0.440811	−	0.897600i	$-0.645309\pi$
$401$	−353.531			−0.881623			−0.440811	+	0.897600i	$0.645309\pi$
$402$			207.907i			0.517183i
$403$			265.644i			0.659166i
$404$		−	784.367i		−	1.94150i
$405$		0			0
$406$			352.031i			0.867073i
$407$	84.3923	−	109.055i	0.207352	−	0.267948i
$408$	−333.846			−0.818250
$409$			625.515i			1.52938i	0.644401	+	0.764688i	$0.277107\pi$
$409$			625.515i			1.52938i	−0.644401	+	0.764688i	$0.722893\pi$
$410$		0			0
$411$	50.6025			0.123121
$412$	1191.29			2.89148
$413$		−	162.166i		−	0.392653i
$414$			151.877i			0.366854i
$415$		0			0
$416$	329.611			0.792335
$417$		−	296.793i		−	0.711735i
$418$	−77.8846	+	100.645i	−0.186327	+	0.240778i
$419$	100.631			0.240169			0.120084	−	0.992764i	$-0.461683\pi$
$419$	100.631			0.240169			0.120084	+	0.992764i	$0.461683\pi$
$420$		0			0
$421$	−25.7616			−0.0611914			−0.0305957	−	0.999532i	$-0.509740\pi$
$421$	−25.7616			−0.0611914			−0.0305957	+	0.999532i	$0.509740\pi$
$422$	109.972			0.260597
$423$	−59.4115			−0.140453
$424$		−	189.952i		−	0.448000i
$425$		0			0
$426$		−	423.568i		−	0.994292i
$427$	55.2257			0.129334
$428$		−	633.961i		−	1.48122i
$429$	−358.119	−	277.131i	−0.834777	−	0.645994i
$430$		0			0
$431$			850.178i			1.97257i	0.165047	+	0.986286i	$0.447222\pi$
$431$			850.178i			1.97257i	−0.165047	+	0.986286i	$0.552778\pi$
$432$	113.196			0.262028
$433$	502.631			1.16081			0.580405	−	0.814328i	$-0.302894\pi$
$433$	502.631			1.16081			0.580405	+	0.814328i	$0.302894\pi$
$434$	101.982			0.234982
$435$		0			0
$436$			1300.37i			2.98250i
$437$			46.9910i			0.107531i
$438$	−244.392			−0.557973
$439$			141.132i			0.321485i	0.986996	+	0.160743i	$0.0513889\pi$
$439$			141.132i			0.321485i	−0.986996	+	0.160743i	$0.948611\pi$
$440$		0			0
$441$	126.962			0.287895
$442$		−	1026.19i		−	2.32171i
$443$	871.538			1.96736			0.983678	−	0.179940i	$-0.0575905\pi$
$443$	871.538			1.96736			0.983678	+	0.179940i	$0.0575905\pi$
$444$	−183.779			−0.413918
$445$		0			0
$446$		−	486.189i		−	1.09011i
$447$		−	284.564i		−	0.636607i
$448$			98.6671i			0.220239i
$449$	−122.364			−0.272526			−0.136263	−	0.990673i	$-0.543509\pi$
$449$	−122.364			−0.272526			−0.136263	+	0.990673i	$0.543509\pi$
$450$		0			0
$451$	−12.0488	−	9.32398i	−0.0267157	−	0.0206740i
$452$	1407.71			3.11440
$453$			397.588i			0.877677i
$454$	1164.47			2.56490
$455$		0			0
$456$	89.4538			0.196171
$457$			581.864i			1.27323i	0.771184	+	0.636613i	$0.219665\pi$
$457$			581.864i			1.27323i	−0.771184	+	0.636613i	$0.780335\pi$
$458$			1056.60i			2.30699i
$459$		−	63.5482i		−	0.138449i
$460$		0			0
$461$			54.4595i			0.118133i	0.998254	+	0.0590667i	$0.0188124\pi$
$461$			54.4595i			0.118133i	−0.998254	+	0.0590667i	$0.981188\pi$
$462$	−106.392	+	137.484i	−0.230286	+	0.297584i
$463$	−356.382			−0.769724			−0.384862	−	0.922974i	$-0.625751\pi$
$463$	−356.382			−0.769724			−0.384862	+	0.922974i	$0.625751\pi$
$464$		−	840.483i		−	1.81139i
$465$		0			0
$466$	−515.854			−1.10698
$467$	−298.613			−0.639428			−0.319714	−	0.947514i	$-0.603587\pi$
$467$	−298.613			−0.639428			−0.319714	+	0.947514i	$0.603587\pi$
$468$			603.504i			1.28954i
$469$			87.8720i			0.187360i
$470$		0			0
$471$	185.090			0.392972
$472$			988.898i			2.09512i
$473$	−208.374	−	161.251i	−0.440538	−	0.340911i
$474$	−597.142			−1.25979
$475$		0			0
$476$	−267.531			−0.562039
$477$	36.1577			0.0758023
$478$	223.874			0.468356
$479$			333.755i			0.696774i	0.937351	+	0.348387i	$0.113270\pi$
$479$			333.755i			0.696774i	−0.937351	+	0.348387i	$0.886730\pi$
$480$		0			0
$481$		−	297.943i		−	0.619424i
$482$	1239.25			2.57106
$483$			64.1909i			0.132901i
$484$	256.962	+	991.396i	0.530912	+	2.04834i
$485$		0			0
$486$			55.0343i			0.113239i
$487$	709.251			1.45637			0.728184	−	0.685382i	$-0.240365\pi$
$487$	709.251			1.45637			0.728184	+	0.685382i	$0.240365\pi$
$488$	−336.771			−0.690104
$489$	204.497			0.418195
$490$		0			0
$491$		−	31.9912i		−	0.0651551i	−0.999469	−	0.0325776i	$-0.989628\pi$
$491$		−	31.9912i		−	0.0651551i	0.999469	−	0.0325776i	$-0.0103716\pi$
$492$			20.3047i			0.0412696i
$493$	−471.846			−0.957091
$494$			274.968i			0.556615i
$495$		0			0
$496$	−243.485			−0.490897
$497$		−	179.021i		−	0.360203i
$498$	−440.238			−0.884013
$499$	27.6922			0.0554954			0.0277477	−	0.999615i	$-0.491167\pi$
$499$	27.6922			0.0554954			0.0277477	+	0.999615i	$0.491167\pi$
$500$		0			0
$501$			2.39891i			0.00478825i
$502$			232.032i			0.462216i
$503$		−	69.6946i		−	0.138558i	−0.997597	−	0.0692789i	$-0.977930\pi$
$503$		−	69.6946i		−	0.138558i	0.997597	−	0.0692789i	$-0.0220698\pi$
$504$	122.196			0.242453
$505$		0			0
$506$	440.414	+	340.815i	0.870383	+	0.673548i
$507$	−685.683			−1.35243
$508$			1610.91i			3.17109i
$509$	−252.732			−0.496527			−0.248263	−	0.968693i	$-0.579860\pi$
$509$	−252.732			−0.496527			−0.248263	+	0.968693i	$0.579860\pi$
$510$		0			0
$511$	−103.292			−0.202138
$512$		−	1071.21i		−	2.09221i
$513$			17.0277i			0.0331924i
$514$			727.916i			1.41618i
$515$		0			0
$516$			351.153i			0.680530i
$517$	−133.321	+	172.282i	−0.257873	+	0.333233i
$518$	−114.382			−0.220815
$519$			117.501i			0.226398i
$520$		0			0
$521$	−611.864			−1.17440			−0.587201	−	0.809441i	$-0.699770\pi$
$521$	−611.864			−1.17440			−0.587201	+	0.809441i	$0.699770\pi$
$522$	408.631			0.782817
$523$		−	6.31868i		−	0.0120816i	−0.999982	−	0.00604080i	$-0.998077\pi$
$523$		−	6.31868i		−	0.0120816i	0.999982	−	0.00604080i	$-0.00192286\pi$
$524$			839.840i			1.60275i
$525$		0			0
$526$	−64.5256			−0.122672
$527$			136.692i			0.259378i
$528$	254.014	−	328.246i	0.481087	−	0.621678i
$529$	−323.372			−0.611289
$530$		0			0
$531$	−188.238			−0.354498
$532$	71.6846			0.134746
$533$	−32.9179			−0.0617597
$534$			655.379i			1.22730i
$535$		0			0
$536$		−	535.850i		−	0.999721i
$537$	−236.603			−0.440601
$538$		−	607.695i		−	1.12955i
$539$	284.904	−	368.163i	0.528578	−	0.683048i
$540$		0			0
$541$		−	427.018i		−	0.789311i	−0.918829	−	0.394656i	$-0.870864\pi$
$541$		−	427.018i		−	0.789311i	0.918829	−	0.394656i	$-0.129136\pi$
$542$	−1503.86			−2.77464
$543$	298.210			0.549190
$544$	169.608			0.311779
$545$		0			0
$546$			375.613i			0.687936i
$547$			783.217i			1.43184i	0.698182	+	0.715921i	$0.253993\pi$
$547$			783.217i			1.43184i	−0.698182	+	0.715921i	$0.746007\pi$
$548$	−247.282			−0.451245
$549$		−	64.1048i		−	0.116767i
$550$		0			0
$551$	126.431			0.229457
$552$		−	391.441i		−	0.709133i
$553$	−252.382			−0.456387
$554$	922.999			1.66606
$555$		0			0
$556$			1450.36i			2.60855i
$557$			508.385i			0.912720i	0.889795	+	0.456360i	$0.150847\pi$
$557$			508.385i			0.912720i	−0.889795	+	0.456360i	$0.849153\pi$
$558$		−	118.379i		−	0.212148i
$559$	−569.290			−1.01841
$560$		0			0
$561$	−184.277	−	142.603i	−0.328479	−	0.254195i
$562$	−1162.20			−2.06797
$563$		−	670.786i		−	1.19145i	−0.803189	−	0.595725i	$-0.796865\pi$
$563$		−	670.786i		−	1.19145i	0.803189	−	0.595725i	$-0.203135\pi$
$564$	290.329			0.514769
$565$		0			0
$566$	816.782			1.44308
$567$			23.2602i			0.0410234i
$568$			1091.68i			1.92198i
$569$			756.359i			1.32928i	0.747165	+	0.664639i	$0.231414\pi$
$569$			756.359i			1.32928i	−0.747165	+	0.664639i	$0.768586\pi$
$570$		0			0
$571$			647.983i			1.13482i	0.823435	+	0.567411i	$0.192055\pi$
$571$			647.983i			1.13482i	−0.823435	+	0.567411i	$0.807945\pi$
$572$	1750.04	+	1354.27i	3.05951	+	2.36761i
$573$	238.617			0.416434
$574$			12.6374i			0.0220163i
$575$		0			0
$576$	114.531			0.198838
$577$	−78.3257			−0.135746			−0.0678732	−	0.997694i	$-0.521621\pi$
$577$	−78.3257			−0.135746			−0.0678732	+	0.997694i	$0.521621\pi$
$578$			492.254i			0.851650i
$579$		−	184.326i		−	0.318352i
$580$		0			0
$581$	−186.067			−0.320252
$582$			1019.64i			1.75196i
$583$	81.1384	−	104.850i	0.139174	−	0.179846i
$584$	629.885			1.07857
$585$		0			0
$586$	705.149			1.20333
$587$	−190.515			−0.324558			−0.162279	−	0.986745i	$-0.551884\pi$
$587$	−190.515			−0.324558			−0.162279	+	0.986745i	$0.551884\pi$
$588$	−620.429			−1.05515
$589$		−	36.6265i		−	0.0621842i
$590$		0			0
$591$			57.8723i			0.0979226i
$592$	273.090			0.461300
$593$			770.010i			1.29850i	0.760575	+	0.649250i	$0.224917\pi$
$593$			770.010i			1.29850i	−0.760575	+	0.649250i	$0.775083\pi$
$594$	159.588	+	123.498i	0.268667	+	0.207909i
$595$		0			0
$596$			1390.59i			2.33321i
$597$	578.603			0.969183
$598$	1203.23			2.01210
$599$	−891.517			−1.48834			−0.744171	−	0.667989i	$-0.767155\pi$
$599$	−891.517			−1.48834			−0.744171	+	0.667989i	$0.767155\pi$
$600$		0			0
$601$			75.5063i			0.125635i	0.998025	+	0.0628173i	$0.0200085\pi$
$601$			75.5063i			0.125635i	−0.998025	+	0.0628173i	$0.979991\pi$
$602$			218.553i			0.363046i
$603$	102.000			0.169154
$604$		−	1942.91i		−	3.21674i
$605$		0			0
$606$	−566.669			−0.935098
$607$			442.017i			0.728200i	0.931360	+	0.364100i	$0.118623\pi$
$607$			442.017i			0.728200i	−0.931360	+	0.364100i	$0.881377\pi$
$608$	−45.4462			−0.0747471
$609$	172.708			0.283592
$610$		0			0
$611$			470.682i			0.770347i
$612$			310.544i			0.507425i
$613$		−	544.346i		−	0.888003i	−0.896026	−	0.444002i	$-0.853558\pi$
$613$		−	544.346i		−	0.888003i	0.896026	−	0.444002i	$-0.146442\pi$
$614$	−1504.99			−2.45113
$615$		0			0
$616$	274.210	−	354.344i	0.445146	−	0.575234i
$617$	−406.946			−0.659556			−0.329778	−	0.944059i	$-0.606974\pi$
$617$	−406.946			−0.659556			−0.329778	+	0.944059i	$0.606974\pi$
$618$		−	860.652i		−	1.39264i
$619$	−218.939			−0.353697			−0.176849	−	0.984238i	$-0.556590\pi$
$619$	−218.939			−0.353697			−0.176849	+	0.984238i	$0.556590\pi$
$620$		0			0
$621$	74.5115			0.119986
$622$		−	1064.83i		−	1.71194i
$623$			276.996i			0.444616i
$624$		−	896.785i		−	1.43716i
$625$		0			0
$626$		−	920.380i		−	1.47026i
$627$	49.3768	+	38.2104i	0.0787509	+	0.0609416i
$628$	−904.487			−1.44027
$629$		−	153.312i		−	0.243740i
$630$		0			0
$631$	185.664			0.294238			0.147119	−	0.989119i	$-0.453000\pi$
$631$	185.664			0.294238			0.147119	+	0.989119i	$0.453000\pi$
$632$	1539.04			2.43520
$633$		−	53.9525i		−	0.0852330i
$634$		−	1592.91i		−	2.51247i
$635$		0			0
$636$	−176.694			−0.277820
$637$		−	1005.84i		−	1.57903i
$638$	916.974	−	1184.95i	1.43726	−	1.85728i
$639$	−207.804			−0.325202
$640$		0			0
$641$	375.618			0.585988			0.292994	−	0.956114i	$-0.405348\pi$
$641$	375.618			0.585988			0.292994	+	0.956114i	$0.405348\pi$
$642$	−458.008			−0.713407
$643$	454.764			0.707253			0.353627	−	0.935387i	$-0.384948\pi$
$643$	454.764			0.707253			0.353627	+	0.935387i	$0.384948\pi$
$644$		−	313.685i		−	0.487089i
$645$		0			0
$646$			141.490i			0.219024i
$647$	−724.273			−1.11943			−0.559716	−	0.828684i	$-0.689090\pi$
$647$	−724.273			−1.11943			−0.559716	+	0.828684i	$0.689090\pi$
$648$		−	141.843i		−	0.218893i
$649$	−422.410	+	545.853i	−0.650863	+	0.841069i
$650$		0			0
$651$		−	50.0327i		−	0.0768552i
$652$	−999.328			−1.53271
$653$	336.865			0.515873			0.257937	−	0.966162i	$-0.416957\pi$
$653$	336.865			0.515873			0.257937	+	0.966162i	$0.416957\pi$
$654$	939.458			1.43648
$655$		0			0
$656$		−	30.1720i		−	0.0459939i
$657$			119.900i			0.182496i
$658$	180.697			0.274616
$659$			41.1163i			0.0623919i	0.999513	+	0.0311960i	$0.00993159\pi$
$659$			41.1163i			0.0623919i	−0.999513	+	0.0311960i	$0.990068\pi$
$660$		0			0
$661$	−831.969			−1.25865			−0.629326	−	0.777141i	$-0.716669\pi$
$661$	−831.969			−1.25865			−0.629326	+	0.777141i	$0.716669\pi$
$662$			810.483i			1.22430i
$663$	−503.454			−0.759357
$664$	1134.65			1.70881
$665$		0			0
$666$			132.772i			0.199358i
$667$		−	553.249i		−	0.829459i
$668$		−	11.7229i		−	0.0175493i
$669$	−238.526			−0.356540
$670$		0			0
$671$	−185.891	−	143.852i	−0.277036	−	0.214385i
$672$	−62.0807			−0.0923820
$673$			682.274i			1.01378i	0.862011	+	0.506890i	$0.169205\pi$
$673$			682.274i			1.01378i	−0.862011	+	0.506890i	$0.830795\pi$
$674$	−628.925			−0.933124
$675$		0			0
$676$	3350.76			4.95675
$677$		−	483.554i		−	0.714261i	−0.934055	−	0.357130i	$-0.883755\pi$
$677$		−	483.554i		−	0.714261i	0.934055	−	0.357130i	$-0.116245\pi$
$678$		−	1017.01i		−	1.50001i
$679$			430.951i			0.634684i
$680$		0			0
$681$		−	571.291i		−	0.838900i
$682$	−343.274	−	265.644i	−0.503335	−	0.389507i
$683$	−875.097			−1.28126			−0.640628	−	0.767852i	$-0.721326\pi$
$683$	−875.097			−1.28126			−0.640628	+	0.767852i	$0.721326\pi$
$684$		−	83.2100i		−	0.121652i
$685$		0			0
$686$	−833.241			−1.21464
$687$	518.372			0.754544
$688$		−	521.801i		−	0.758432i
$689$		−	286.455i		−	0.415755i
$690$		0			0
$691$	845.615			1.22376			0.611878	−	0.790952i	$-0.290414\pi$
$691$	845.615			1.22376			0.611878	+	0.790952i	$0.290414\pi$
$692$		−	574.197i		−	0.829764i
$693$	67.4500	+	52.1964i	0.0973305	+	0.0753194i
$694$	−2029.44			−2.92427
$695$		0			0
$696$	−1053.18			−1.51320
$697$	−16.9385			−0.0243020
$698$	−644.729			−0.923681
$699$			253.079i			0.362059i
$700$		0			0
$701$		−	842.982i		−	1.20254i	−0.799045	−	0.601271i	$-0.794661\pi$
$701$		−	842.982i		−	1.20254i	0.799045	−	0.601271i	$-0.205339\pi$
$702$	436.004			0.621088
$703$			41.0799i			0.0584351i
$704$	257.009	−	332.116i	0.365070	−	0.471756i
$705$		0			0
$706$			691.932i			0.980074i
$707$	−239.503			−0.338759
$708$	919.874			1.29926
$709$	1227.03			1.73065			0.865325	−	0.501212i	$-0.167112\pi$
$709$	1227.03			1.73065			0.865325	+	0.501212i	$0.167112\pi$
$710$		0			0
$711$			292.960i			0.412039i
$712$		−	1689.14i		−	2.37239i
$713$	−160.274			−0.224788
$714$			193.279i			0.270698i
$715$		0			0
$716$	1156.22			1.61483
$717$		−	109.833i		−	0.153185i
$718$	2496.40			3.47688
$719$	343.986			0.478423			0.239211	−	0.970968i	$-0.423111\pi$
$719$	343.986			0.478423			0.239211	+	0.970968i	$0.423111\pi$
$720$		0			0
$721$		−	363.754i		−	0.504514i
$722$			1236.58i			1.71272i
$723$		−	607.980i		−	0.840913i
$724$	−1457.28			−2.01282
$725$		0			0
$726$	716.238	−	185.643i	0.986554	−	0.255706i
$727$	−205.138			−0.282171			−0.141086	−	0.989997i	$-0.545059\pi$
$727$	−205.138			−0.282171			−0.141086	+	0.989997i	$0.545059\pi$
$728$		−	968.086i		−	1.32979i
$729$	27.0000			0.0370370
$730$		0			0
$731$	−292.939			−0.400737
$732$			313.264i			0.427957i
$733$		−	849.486i		−	1.15892i	−0.815002	−	0.579458i	$-0.803264\pi$
$733$		−	849.486i		−	1.15892i	0.815002	−	0.579458i	$-0.196736\pi$
$734$		−	1268.90i		−	1.72875i
$735$		0			0
$736$			198.868i			0.270202i
$737$	228.890	−	295.779i	0.310570	−	0.401329i
$738$	14.6692			0.0198769
$739$			1206.77i			1.63298i	0.577360	+	0.816490i	$0.304083\pi$
$739$			1206.77i			1.63298i	−0.577360	+	0.816490i	$0.695917\pi$
$740$		0			0
$741$	134.900			0.182051
$742$	−109.972			−0.148210
$743$		−	229.932i		−	0.309464i	−0.987956	−	0.154732i	$-0.950549\pi$
$743$		−	229.932i		−	0.309464i	0.987956	−	0.154732i	$-0.0494514\pi$
$744$			305.103i			0.410085i
$745$		0			0
$746$	1284.39			1.72171
$747$			215.982i			0.289133i
$748$	900.515	+	696.866i	1.20390	+	0.931639i
$749$	−193.577			−0.258447
$750$		0			0
$751$	616.269			0.820598			0.410299	−	0.911951i	$-0.365424\pi$
$751$	616.269			0.820598			0.410299	+	0.911951i	$0.365424\pi$
$752$	−431.419			−0.573696
$753$	113.836			0.151176
$754$		−	3237.33i		−	4.29355i
$755$		0			0
$756$		−	113.667i		−	0.150353i
$757$	−85.4181			−0.112838			−0.0564188	−	0.998407i	$-0.517968\pi$
$757$	−85.4181			−0.112838			−0.0564188	+	0.998407i	$0.517968\pi$
$758$		−	939.263i		−	1.23913i
$759$	167.205	−	216.068i	0.220297	−	0.284675i
$760$		0			0
$761$			439.940i			0.578108i	0.957313	+	0.289054i	$0.0933406\pi$
$761$			439.940i			0.578108i	−0.957313	+	0.289054i	$0.906659\pi$
$762$	1163.81			1.52731
$763$	397.061			0.520395
$764$	−1166.06			−1.52626
$765$		0			0
$766$		−	1169.49i		−	1.52675i
$767$			1491.30i			1.94433i
$768$	−898.899			−1.17044
$769$		−	928.726i		−	1.20771i	−0.797095	−	0.603853i	$-0.793631\pi$
$769$		−	928.726i		−	1.20771i	0.797095	−	0.603853i	$-0.206369\pi$
$770$		0			0
$771$	357.118			0.463188
$772$			900.754i			1.16678i
$773$	−929.734			−1.20276			−0.601381	−	0.798963i	$-0.705382\pi$
$773$	−929.734			−1.20276			−0.601381	+	0.798963i	$0.705382\pi$
$774$	253.692			0.327768
$775$		0			0
$776$		−	2627.97i		−	3.38656i
$777$			56.1161i			0.0722215i
$778$			1486.42i			1.91056i
$779$	4.53866			0.00582627
$780$		0			0
$781$	−466.315	+	602.589i	−0.597075	+	0.771561i
$782$	619.146			0.791747
$783$		−	200.475i		−	0.256035i
$784$	921.936			1.17594
$785$		0			0
$786$	606.746			0.771942
$787$			250.644i			0.318480i	0.987240	+	0.159240i	$0.0509044\pi$
$787$			250.644i			0.318480i	−0.987240	+	0.159240i	$0.949096\pi$
$788$		−	282.807i		−	0.358893i
$789$			31.6564i			0.0401222i
$790$		0			0
$791$		−	429.837i		−	0.543410i
$792$	−411.315	−	318.297i	−0.519338	−	0.401891i
$793$	−507.864			−0.640434
$794$			1489.14i			1.87549i
$795$		0			0
$796$	−2827.49			−3.55212
$797$	−872.319			−1.09450			−0.547252	−	0.836968i	$-0.684326\pi$
$797$	−872.319			−1.09450			−0.547252	+	0.836968i	$0.684326\pi$
$798$		−	51.7889i		−	0.0648983i
$799$			242.198i			0.303127i
$800$		0			0
$801$	321.531			0.401412
$802$		−	1248.12i		−	1.55626i
$803$	347.685	+	269.057i	0.432982	+	0.335064i
$804$	−498.449			−0.619961
$805$		0			0
$806$	−937.843			−1.16358
$807$	−298.137			−0.369439
$808$	1460.50			1.80756
$809$			963.696i			1.19122i	0.803274	+	0.595609i	$0.203089\pi$
$809$			963.696i			1.19122i	−0.803274	+	0.595609i	$0.796911\pi$
$810$		0			0
$811$		−	583.285i		−	0.719217i	−0.933103	−	0.359609i	$-0.882910\pi$
$811$		−	583.285i		−	0.719217i	0.933103	−	0.359609i	$-0.117090\pi$
$812$	−843.979			−1.03938
$813$			737.797i			0.907499i
$814$	385.013	+	297.943i	0.472989	+	0.366023i
$815$		0			0
$816$		−	461.457i		−	0.565511i
$817$	78.4926			0.0960742
$818$	−2208.35			−2.69970
$819$	184.277			0.225002
$820$		0			0
$821$			211.927i			0.258132i	0.991636	+	0.129066i	$0.0411980\pi$
$821$			211.927i			0.258132i	−0.991636	+	0.129066i	$0.958802\pi$
$822$			178.650i			0.217336i
$823$	1049.19			1.27484			0.637421	−	0.770516i	$-0.280001\pi$
$823$	1049.19			1.27484			0.637421	+	0.770516i	$0.280001\pi$
$824$			2218.20i			2.69199i
$825$		0			0
$826$	572.518			0.693121
$827$			1226.67i			1.48328i	0.670800	+	0.741638i	$0.265951\pi$
$827$			1226.67i			1.48328i	−0.670800	+	0.741638i	$0.734049\pi$
$828$	−364.119			−0.439757
$829$	579.395			0.698908			0.349454	−	0.936954i	$-0.386367\pi$
$829$	579.395			0.698908			0.349454	+	0.936954i	$0.386367\pi$
$830$		0			0
$831$		−	452.826i		−	0.544917i
$832$		−	907.358i		−	1.09057i
$833$		−	517.573i		−	0.621337i
$834$	1047.82			1.25637
$835$		0			0
$836$	−241.292	−	186.725i	−0.288627	−	0.223355i
$837$	−58.0770			−0.0693870
$838$			355.272i			0.423952i
$839$	−1198.03			−1.42793			−0.713964	−	0.700182i	$-0.753102\pi$
$839$	−1198.03			−1.42793			−0.713964	+	0.700182i	$0.753102\pi$
$840$		0			0
$841$	−647.533			−0.769956
$842$		−	90.9501i		−	0.108017i
$843$			570.177i			0.676367i
$844$			263.652i			0.312384i
$845$		0			0
$846$		−	209.750i		−	0.247931i
$847$	302.718	−	78.4619i	0.357400	−	0.0926351i
$848$	262.560			0.309623
$849$		−	400.716i		−	0.471985i
$850$		0			0
$851$	179.762			0.211236
$852$	1015.49			1.19188
$853$		−	1015.06i		−	1.18999i	−0.803728	−	0.594997i	$-0.797153\pi$
$853$		−	1015.06i		−	1.18999i	0.803728	−	0.594997i	$-0.202847\pi$
$854$			194.972i			0.228304i
$855$		0			0
$856$	1180.45			1.37903
$857$		−	1574.17i		−	1.83684i	−0.395603	−	0.918422i	$-0.629464\pi$
$857$		−	1574.17i		−	1.83684i	0.395603	−	0.918422i	$-0.370536\pi$
$858$	978.400	−	1264.32i	1.14033	−	1.47357i
$859$	99.9230			0.116325			0.0581624	−	0.998307i	$-0.481476\pi$
$859$	99.9230			0.116325			0.0581624	+	0.998307i	$0.481476\pi$
$860$		0			0
$861$	6.19993			0.00720084
$862$	−3001.51			−3.48204
$863$	1405.55			1.62868			0.814339	−	0.580389i	$-0.197100\pi$
$863$	1405.55			1.62868			0.814339	+	0.580389i	$0.197100\pi$
$864$			72.0620i			0.0834051i
$865$		0			0
$866$			1774.51i			2.04909i
$867$	241.501			0.278548
$868$			244.498i			0.281679i
$869$	849.524	+	657.407i	0.977588	+	0.756509i
$870$		0			0
$871$		−	808.085i		−	0.927766i
$872$	−2421.31			−2.77673
$873$	500.238			0.573011
$874$	−165.900			−0.189817
$875$		0			0
$876$		−	585.919i		−	0.668858i
$877$			1231.05i			1.40370i	0.712324	+	0.701851i	$0.247643\pi$
$877$			1231.05i			1.40370i	−0.712324	+	0.701851i	$0.752357\pi$
$878$	−498.260			−0.567494
$879$		−	345.948i		−	0.393570i
$880$		0			0
$881$	419.359			0.476003			0.238002	−	0.971265i	$-0.423508\pi$
$881$	419.359			0.476003			0.238002	+	0.971265i	$0.423508\pi$
$882$			448.232i			0.508199i
$883$	−867.856			−0.982850			−0.491425	−	0.870920i	$-0.663524\pi$
$883$	−867.856			−0.982850			−0.491425	+	0.870920i	$0.663524\pi$
$884$	2460.25			2.78309
$885$		0			0
$886$			3076.93i			3.47283i
$887$		−	484.124i		−	0.545800i	−0.962042	−	0.272900i	$-0.912017\pi$
$887$		−	484.124i		−	0.545800i	0.962042	−	0.272900i	$-0.0879828\pi$
$888$		−	342.201i		−	0.385361i
$889$	491.885			0.553301
$890$		0			0
$891$	60.5885	−	78.2946i	0.0680005	−	0.0878727i
$892$	1165.62			1.30674
$893$		−	64.8968i		−	0.0726728i
$894$	1004.64			1.12376
$895$		0			0
$896$	−491.709			−0.548782
$897$		−	590.310i		−	0.658093i
$898$		−	432.001i		−	0.481070i
$899$			431.222i			0.479669i
$900$		0			0
$901$		−	147.401i		−	0.163597i
$902$	32.9179	−	42.5377i	0.0364943	−	0.0471593i
$903$	107.223			0.118741
$904$			2621.18i			2.89954i
$905$		0			0
$906$	−1403.67			−1.54930
$907$	794.610			0.876086			0.438043	−	0.898954i	$-0.355672\pi$
$907$	794.610			0.876086			0.438043	+	0.898954i	$0.355672\pi$
$908$			2791.76i			3.07462i
$909$			278.010i			0.305841i
$910$		0			0
$911$	256.042			0.281056			0.140528	−	0.990077i	$-0.455120\pi$
$911$	256.042			0.281056			0.140528	+	0.990077i	$0.455120\pi$
$912$			123.647i			0.135578i
$913$	626.305	+	484.668i	0.685986	+	0.530852i
$914$	−2054.24			−2.24753
$915$		0			0
$916$	−2533.15			−2.76545
$917$	256.441			0.279652
$918$	224.354			0.244394
$919$		−	546.682i		−	0.594866i	−0.954743	−	0.297433i	$-0.903870\pi$
$919$		−	546.682i		−	0.594866i	0.954743	−	0.297433i	$-0.0961305\pi$
$920$		0			0
$921$			738.354i			0.801687i
$922$	−192.267			−0.208532
$923$			1646.30i			1.78365i
$924$	−329.611	−	255.071i	−0.356722	−	0.276051i
$925$		0			0
$926$		−	1258.19i		−	1.35874i
$927$	−422.238			−0.455489
$928$	535.061			0.576575
$929$	1434.55			1.54419			0.772096	−	0.635506i	$-0.219209\pi$
$929$	1434.55			1.54419			0.772096	+	0.635506i	$0.219209\pi$
$930$		0			0
$931$			138.683i			0.148962i
$932$		−	1236.74i		−	1.32697i
$933$	−522.406			−0.559921
$934$		−	1054.24i		−	1.12874i
$935$		0			0
$936$	−1123.73			−1.20057
$937$			377.432i			0.402809i	0.979508	+	0.201405i	$0.0645506\pi$
$937$			377.432i			0.402809i	−0.979508	+	0.201405i	$0.935449\pi$
$938$	−310.228			−0.330734
$939$	−451.541			−0.480874
$940$		0			0
$941$		−	721.127i		−	0.766341i	−0.923678	−	0.383171i	$-0.874832\pi$
$941$		−	721.127i		−	0.766341i	0.923678	−	0.383171i	$-0.125168\pi$
$942$			653.450i			0.693684i
$943$		−	19.8607i		−	0.0210612i
$944$	−1366.90			−1.44799
$945$		0			0
$946$	569.290	−	735.656i	0.601786	−	0.777649i
$947$	1526.20			1.61162			0.805808	−	0.592177i	$-0.201731\pi$
$947$	1526.20			1.61162			0.805808	+	0.592177i	$0.201731\pi$
$948$		−	1431.62i		−	1.51015i
$949$	949.892			1.00094
$950$		0			0
$951$	−781.486			−0.821752
$952$		−	498.147i		−	0.523264i
$953$			252.264i			0.264705i	0.991203	+	0.132353i	$0.0422532\pi$
$953$			252.264i			0.264705i	−0.991203	+	0.132353i	$0.957747\pi$
$954$			127.653i			0.133808i
$955$		0			0
$956$			536.728i			0.561431i
$957$	−581.338	−	449.870i	−0.607459	−	0.470084i
$958$	−1178.31			−1.22996
$959$			75.5063i			0.0787345i
$960$		0			0
$961$	−836.077			−0.870007
$962$	1051.87			1.09342
$963$			224.700i			0.233333i
$964$			2971.05i			3.08200i
$965$		0			0
$966$	−226.623			−0.234600
$967$		−	39.6815i		−	0.0410357i	−0.999789	−	0.0205179i	$-0.993469\pi$
$967$		−	39.6815i		−	0.0410357i	0.999789	−	0.0205179i	$-0.00653149\pi$
$968$	−1846.00	+	478.467i	−1.90702	+	0.494284i
$969$	69.4153			0.0716360
$970$		0			0
$971$	−790.585			−0.814196			−0.407098	−	0.913384i	$-0.633459\pi$
$971$	−790.585			−0.814196			−0.407098	+	0.913384i	$0.633459\pi$
$972$	−131.942			−0.135743
$973$	442.859			0.455148
$974$			2503.98i			2.57082i
$975$		0			0
$976$		−	465.500i		−	0.476946i
$977$	309.474			0.316760			0.158380	−	0.987378i	$-0.449373\pi$
$977$	309.474			0.316760			0.158380	+	0.987378i	$0.449373\pi$
$978$			721.969i			0.738209i
$979$	721.520	−	932.374i	0.736997	−	0.952374i
$980$		0			0
$981$		−	460.901i		−	0.469827i
$982$	112.943			0.115014
$983$	−529.604			−0.538763			−0.269381	−	0.963034i	$-0.586819\pi$
$983$	−529.604			−0.538763			−0.269381	+	0.963034i	$0.586819\pi$
$984$	−37.8076			−0.0384224
$985$		0			0
$986$		−	1665.83i		−	1.68948i
$987$		−	88.6506i		−	0.0898183i
$988$	−659.223			−0.667230
$989$		−	343.476i		−	0.347297i
$990$		0			0
$991$	1038.56			1.04799			0.523997	−	0.851720i	$-0.324440\pi$
$991$	1038.56			1.04799			0.523997	+	0.851720i	$0.324440\pi$
$992$		−	155.005i		−	0.156255i
$993$	397.626			0.400429
$994$	632.025			0.635840
$995$		0			0
$996$		−	1055.45i		−	1.05969i
$997$		−	1105.88i		−	1.10921i	−0.832115	−	0.554603i	$-0.812870\pi$
$997$		−	1105.88i		−	1.10921i	0.832115	−	0.554603i	$-0.187130\pi$
$998$			97.7660i			0.0979619i
$999$	65.1384			0.0652036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.3.b.a.76.4		4
5.2	odd	4		825.3.h.a.274.1		8
5.3	odd	4		825.3.h.a.274.8		8
5.4	even	2		33.3.c.a.10.1	✓	4
11.10	odd	2	inner	825.3.b.a.76.1		4
15.14	odd	2		99.3.c.b.10.4		4
20.19	odd	2		528.3.j.c.241.4		4
40.19	odd	2		2112.3.j.d.769.2		4
40.29	even	2		2112.3.j.a.769.3		4
55.4	even	10		363.3.g.e.94.4		16
55.9	even	10		363.3.g.e.40.1		16
55.14	even	10		363.3.g.e.112.1		16
55.19	odd	10		363.3.g.e.112.4		16
55.24	odd	10		363.3.g.e.40.4		16
55.29	odd	10		363.3.g.e.94.1		16
55.32	even	4		825.3.h.a.274.7		8
55.39	odd	10		363.3.g.e.118.1		16
55.43	even	4		825.3.h.a.274.2		8
55.49	even	10		363.3.g.e.118.4		16
55.54	odd	2		33.3.c.a.10.4	yes	4
60.59	even	2		1584.3.j.f.1297.2		4
165.164	even	2		99.3.c.b.10.1		4
220.219	even	2		528.3.j.c.241.3		4
440.109	odd	2		2112.3.j.a.769.4		4
440.219	even	2		2112.3.j.d.769.1		4
660.659	odd	2		1584.3.j.f.1297.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.c.a.10.1	✓	4	5.4	even	2
33.3.c.a.10.4	yes	4	55.54	odd	2
99.3.c.b.10.1		4	165.164	even	2
99.3.c.b.10.4		4	15.14	odd	2
363.3.g.e.40.1		16	55.9	even	10
363.3.g.e.40.4		16	55.24	odd	10
363.3.g.e.94.1		16	55.29	odd	10
363.3.g.e.94.4		16	55.4	even	10
363.3.g.e.112.1		16	55.14	even	10
363.3.g.e.112.4		16	55.19	odd	10
363.3.g.e.118.1		16	55.39	odd	10
363.3.g.e.118.4		16	55.49	even	10
528.3.j.c.241.3		4	220.219	even	2
528.3.j.c.241.4		4	20.19	odd	2
825.3.b.a.76.1		4	11.10	odd	2	inner
825.3.b.a.76.4		4	1.1	even	1	trivial
825.3.h.a.274.1		8	5.2	odd	4
825.3.h.a.274.2		8	55.43	even	4
825.3.h.a.274.7		8	55.32	even	4
825.3.h.a.274.8		8	5.3	odd	4
1584.3.j.f.1297.1		4	660.659	odd	2
1584.3.j.f.1297.2		4	60.59	even	2
2112.3.j.a.769.3		4	40.29	even	2
2112.3.j.a.769.4		4	440.109	odd	2
2112.3.j.d.769.1		4	440.219	even	2
2112.3.j.d.769.2		4	40.19	odd	2

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	825.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.53045i q^{2} +1.73205 q^{3} -8.46410 q^{4} +6.11492i q^{6} +2.58447i q^{7} -15.7603i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+3.53045i q^{2} +1.73205 q^{3} -8.46410 q^{4} +6.11492i q^{6} +2.58447i q^{7} -15.7603i q^{8} +3.00000 q^{9} +(6.73205 - 8.69940i) q^{11} -14.6603 q^{12} -23.7672i q^{13} -9.12436 q^{14} +21.7846 q^{16} -12.2298i q^{17} +10.5914i q^{18} +3.27698i q^{19} +4.47644i q^{21} +(30.7128 + 23.7672i) q^{22} +14.3397 q^{23} -27.2976i q^{24} +83.9090 q^{26} +5.19615 q^{27} -21.8752i q^{28} -38.5815i q^{29} -11.1769 q^{31} +13.8683i q^{32} +(11.6603 - 15.0678i) q^{33} +43.1769 q^{34} -25.3923 q^{36} +12.5359 q^{37} -11.5692 q^{38} -41.1660i q^{39} -1.38501i q^{41} -15.8038 q^{42} -23.9527i q^{43} +(-56.9808 + 73.6326i) q^{44} +50.6258i q^{46} -19.8038 q^{47} +37.7321 q^{48} +42.3205 q^{49} -21.1827i q^{51} +201.168i q^{52} +12.0526 q^{53} +18.3448i q^{54} +40.7321 q^{56} +5.67589i q^{57} +136.210 q^{58} -62.7461 q^{59} -21.3683i q^{61} -39.4596i q^{62} +7.75341i q^{63} +38.1769 q^{64} +(53.1962 + 41.1660i) q^{66} +34.0000 q^{67} +103.515i q^{68} +24.8372 q^{69} -69.2679 q^{71} -47.2809i q^{72} +39.9665i q^{73} +44.2574i q^{74} -27.7367i q^{76} +(22.4833 + 17.3988i) q^{77} +145.335 q^{78} +97.6532i q^{79} +9.00000 q^{81} +4.88973 q^{82} +71.9941i q^{83} -37.8890i q^{84} +84.5641 q^{86} -66.8251i q^{87} +(-137.105 - 106.099i) q^{88} +107.177 q^{89} +61.4256 q^{91} -121.373 q^{92} -19.3590 q^{93} -69.9166i q^{94} +24.0207i q^{96} +166.746 q^{97} +149.411i q^{98} +(20.1962 - 26.0982i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} + 12 q^{9}+O(q^{10}) \) 4 * q - 20 * q^4 + 12 * q^9	\( 4 q - 20 q^{4} + 12 q^{9} + 20 q^{11} - 24 q^{12} + 12 q^{14} + 4 q^{16} + 12 q^{22} + 92 q^{23} + 204 q^{26} + 80 q^{31} + 12 q^{33} + 48 q^{34} - 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42} - 124 q^{44} - 100 q^{47} + 144 q^{48} + 100 q^{49} - 28 q^{53} + 156 q^{56} + 240 q^{58} + 40 q^{59} + 28 q^{64} + 192 q^{66} + 136 q^{67} - 60 q^{69} - 284 q^{71} + 180 q^{77} + 228 q^{78} + 36 q^{81} - 216 q^{82} - 216 q^{86} - 396 q^{88} + 304 q^{89} + 24 q^{91} - 340 q^{92} - 216 q^{93} + 376 q^{97} + 60 q^{99}+O(q^{100}) \) 4 * q - 20 * q^4 + 12 * q^9 + 20 * q^11 - 24 * q^12 + 12 * q^14 + 4 * q^16 + 12 * q^22 + 92 * q^23 + 204 * q^26 + 80 * q^31 + 12 * q^33 + 48 * q^34 - 60 * q^36 + 64 * q^37 + 120 * q^38 - 84 * q^42 - 124 * q^44 - 100 * q^47 + 144 * q^48 + 100 * q^49 - 28 * q^53 + 156 * q^56 + 240 * q^58 + 40 * q^59 + 28 * q^64 + 192 * q^66 + 136 * q^67 - 60 * q^69 - 284 * q^71 + 180 * q^77 + 228 * q^78 + 36 * q^81 - 216 * q^82 - 216 * q^86 - 396 * q^88 + 304 * q^89 + 24 * q^91 - 340 * q^92 - 216 * q^93 + 376 * q^97 + 60 * q^99

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