LMFDB - Embedded newform 1850.2.c.g.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1849,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1850.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$-1.79129i$ of defining polynomial
Character	$\chi$	$=$	1850.1849
Dual form			1850.2.c.g.1849.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.79129i q^{3} +1.00000 q^{4} +1.79129i q^{6} -2.00000i q^{7} -1.00000 q^{8} -0.208712 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.79129i q^{3} +1.00000 q^{4} +1.79129i q^{6} -2.00000i q^{7} -1.00000 q^{8} -0.208712 q^{9} -0.791288 q^{11} -1.79129i q^{12} +3.79129 q^{13} +2.00000i q^{14} +1.00000 q^{16} +7.58258 q^{17} +0.208712 q^{18} +1.58258i q^{19} -3.58258 q^{21} +0.791288 q^{22} +3.79129 q^{23} +1.79129i q^{24} -3.79129 q^{26} -5.00000i q^{27} -2.00000i q^{28} +3.79129i q^{29} +8.37386i q^{31} -1.00000 q^{32} +1.41742i q^{33} -7.58258 q^{34} -0.208712 q^{36} +(-4.58258 + 4.00000i) q^{37} -1.58258i q^{38} -6.79129i q^{39} +9.79129 q^{41} +3.58258 q^{42} +6.00000 q^{43} -0.791288 q^{44} -3.79129 q^{46} -7.58258i q^{47} -1.79129i q^{48} +3.00000 q^{49} -13.5826i q^{51} +3.79129 q^{52} +1.58258i q^{53} +5.00000i q^{54} +2.00000i q^{56} +2.83485 q^{57} -3.79129i q^{58} +1.58258i q^{59} -12.7913i q^{61} -8.37386i q^{62} +0.417424i q^{63} +1.00000 q^{64} -1.41742i q^{66} +6.37386i q^{67} +7.58258 q^{68} -6.79129i q^{69} -9.16515 q^{71} +0.208712 q^{72} +4.37386i q^{73} +(4.58258 - 4.00000i) q^{74} +1.58258i q^{76} +1.58258i q^{77} +6.79129i q^{78} +8.20871i q^{79} -9.58258 q^{81} -9.79129 q^{82} -15.1652i q^{83} -3.58258 q^{84} -6.00000 q^{86} +6.79129 q^{87} +0.791288 q^{88} +6.00000i q^{89} -7.58258i q^{91} +3.79129 q^{92} +15.0000 q^{93} +7.58258i q^{94} +1.79129i q^{96} -13.5826 q^{97} -3.00000 q^{98} +0.165151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 10 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 10 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 10 q^{9} + 6 q^{11} + 6 q^{13} + 4 q^{16} + 12 q^{17} + 10 q^{18} + 4 q^{21} - 6 q^{22} + 6 q^{23} - 6 q^{26} - 4 q^{32} - 12 q^{34} - 10 q^{36} + 30 q^{41} - 4 q^{42} + 24 q^{43} + 6 q^{44} - 6 q^{46} + 12 q^{49} + 6 q^{52} + 48 q^{57} + 4 q^{64} + 12 q^{68} + 10 q^{72} - 20 q^{81} - 30 q^{82} + 4 q^{84} - 24 q^{86} + 18 q^{87} - 6 q^{88} + 6 q^{92} + 60 q^{93} - 36 q^{97} - 12 q^{98} - 36 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 10 * q^9 + 6 * q^11 + 6 * q^13 + 4 * q^16 + 12 * q^17 + 10 * q^18 + 4 * q^21 - 6 * q^22 + 6 * q^23 - 6 * q^26 - 4 * q^32 - 12 * q^34 - 10 * q^36 + 30 * q^41 - 4 * q^42 + 24 * q^43 + 6 * q^44 - 6 * q^46 + 12 * q^49 + 6 * q^52 + 48 * q^57 + 4 * q^64 + 12 * q^68 + 10 * q^72 - 20 * q^81 - 30 * q^82 + 4 * q^84 - 24 * q^86 + 18 * q^87 - 6 * q^88 + 6 * q^92 + 60 * q^93 - 36 * q^97 - 12 * q^98 - 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	1.79129i		−	1.03420i	−0.855925	−	0.517100i	$-0.827011\pi$
$3$		−	1.79129i		−	1.03420i	0.855925	−	0.517100i	$-0.172989\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			1.79129i			0.731290i
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	−1.00000			−0.353553
$9$	−0.208712			−0.0695707
$10$		0			0
$11$	−0.791288			−0.238582			−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288			−0.238582			−0.119291	+	0.992859i	$0.538062\pi$
$12$		−	1.79129i		−	0.517100i
$13$	3.79129			1.05151			0.525757	−	0.850635i	$-0.323782\pi$
$13$	3.79129			1.05151			0.525757	+	0.850635i	$0.323782\pi$
$14$			2.00000i			0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	7.58258			1.83904			0.919522	−	0.393038i	$-0.128576\pi$
$17$	7.58258			1.83904			0.919522	+	0.393038i	$0.128576\pi$
$18$	0.208712			0.0491939
$19$			1.58258i			0.363068i	0.983385	+	0.181534i	$0.0581062\pi$
$19$			1.58258i			0.363068i	−0.983385	+	0.181534i	$0.941894\pi$
$20$		0			0
$21$	−3.58258			−0.781782
$22$	0.791288			0.168703
$23$	3.79129			0.790538			0.395269	−	0.918565i	$-0.370651\pi$
$23$	3.79129			0.790538			0.395269	+	0.918565i	$0.370651\pi$
$24$			1.79129i			0.365645i
$25$		0			0
$26$	−3.79129			−0.743533
$27$		−	5.00000i		−	0.962250i
$28$		−	2.00000i		−	0.377964i
$29$			3.79129i			0.704024i	0.935995	+	0.352012i	$0.114502\pi$
$29$			3.79129i			0.704024i	−0.935995	+	0.352012i	$0.885498\pi$
$30$		0			0
$31$			8.37386i			1.50399i	0.659169	+	0.751995i	$0.270908\pi$
$31$			8.37386i			1.50399i	−0.659169	+	0.751995i	$0.729092\pi$
$32$	−1.00000			−0.176777
$33$			1.41742i			0.246742i
$34$	−7.58258			−1.30040
$35$		0			0
$36$	−0.208712			−0.0347854
$37$	−4.58258	+	4.00000i	−0.753371	+	0.657596i
$37$	−4.58258	+	4.00000i	−0.753371	+	0.657596i
$38$		−	1.58258i		−	0.256728i
$39$		−	6.79129i		−	1.08748i
$40$		0			0
$41$	9.79129			1.52914			0.764571	−	0.644539i	$-0.222951\pi$
$41$	9.79129			1.52914			0.764571	+	0.644539i	$0.222951\pi$
$42$	3.58258			0.552803
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$	−0.791288			−0.119291
$45$		0			0
$46$	−3.79129			−0.558995
$47$		−	7.58258i		−	1.10603i	−0.833171	−	0.553016i	$-0.813477\pi$
$47$		−	7.58258i		−	1.10603i	0.833171	−	0.553016i	$-0.186523\pi$
$48$		−	1.79129i		−	0.258550i
$49$	3.00000			0.428571
$50$		0			0
$51$		−	13.5826i		−	1.90194i
$52$	3.79129			0.525757
$53$			1.58258i			0.217383i	0.994076	+	0.108692i	$0.0346661\pi$
$53$			1.58258i			0.217383i	−0.994076	+	0.108692i	$0.965334\pi$
$54$			5.00000i			0.680414i
$55$		0			0
$56$			2.00000i			0.267261i
$57$	2.83485			0.375485
$58$		−	3.79129i		−	0.497820i
$59$			1.58258i			0.206034i	0.994680	+	0.103017i	$0.0328496\pi$
$59$			1.58258i			0.206034i	−0.994680	+	0.103017i	$0.967150\pi$
$60$		0			0
$61$		−	12.7913i		−	1.63776i	−0.573967	−	0.818878i	$-0.694596\pi$
$61$		−	12.7913i		−	1.63776i	0.573967	−	0.818878i	$-0.305404\pi$
$62$		−	8.37386i		−	1.06348i
$63$			0.417424i			0.0525905i
$64$	1.00000			0.125000
$65$		0			0
$66$		−	1.41742i		−	0.174473i
$67$			6.37386i			0.778691i	0.921092	+	0.389346i	$0.127299\pi$
$67$			6.37386i			0.778691i	−0.921092	+	0.389346i	$0.872701\pi$
$68$	7.58258			0.919522
$69$		−	6.79129i		−	0.817575i
$70$		0			0
$71$	−9.16515			−1.08770			−0.543852	−	0.839181i	$-0.683035\pi$
$71$	−9.16515			−1.08770			−0.543852	+	0.839181i	$0.683035\pi$
$72$	0.208712			0.0245970
$73$			4.37386i			0.511922i	0.966687	+	0.255961i	$0.0823919\pi$
$73$			4.37386i			0.511922i	−0.966687	+	0.255961i	$0.917608\pi$
$74$	4.58258	−	4.00000i	0.532714	−	0.464991i
$75$		0			0
$76$			1.58258i			0.181534i
$77$			1.58258i			0.180351i
$78$			6.79129i			0.768962i
$79$			8.20871i			0.923552i	0.886997	+	0.461776i	$0.152788\pi$
$79$			8.20871i			0.923552i	−0.886997	+	0.461776i	$0.847212\pi$
$80$		0			0
$81$	−9.58258			−1.06473
$82$	−9.79129			−1.08127
$83$		−	15.1652i		−	1.66459i	−0.554332	−	0.832296i	$-0.687026\pi$
$83$		−	15.1652i		−	1.66459i	0.554332	−	0.832296i	$-0.312974\pi$
$84$	−3.58258			−0.390891
$85$		0			0
$86$	−6.00000			−0.646997
$87$	6.79129			0.728102
$88$	0.791288			0.0843516
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$		−	7.58258i		−	0.794870i
$92$	3.79129			0.395269
$93$	15.0000			1.55543
$94$			7.58258i			0.782083i
$95$		0			0
$96$			1.79129i			0.182823i
$97$	−13.5826			−1.37910			−0.689551	−	0.724237i	$-0.742192\pi$
$97$	−13.5826			−1.37910			−0.689551	+	0.724237i	$0.742192\pi$
$98$	−3.00000			−0.303046
$99$	0.165151			0.0165983
$100$		0			0
$101$	7.58258			0.754494			0.377247	−	0.926113i	$-0.376871\pi$
$101$	7.58258			0.754494			0.377247	+	0.926113i	$0.376871\pi$
$102$			13.5826i			1.34488i
$103$	−6.79129			−0.669165			−0.334583	−	0.942366i	$-0.608595\pi$
$103$	−6.79129			−0.669165			−0.334583	+	0.942366i	$0.608595\pi$
$104$	−3.79129			−0.371766
$105$		0			0
$106$		−	1.58258i		−	0.153713i
$107$		−	5.37386i		−	0.519511i	−0.965674	−	0.259755i	$-0.916358\pi$
$107$		−	5.37386i		−	0.519511i	0.965674	−	0.259755i	$-0.0836420\pi$
$108$		−	5.00000i		−	0.481125i
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$	7.16515	+	8.20871i	0.680086	+	0.779136i
$112$		−	2.00000i		−	0.188982i
$113$	10.4174			0.979989			0.489994	−	0.871726i	$-0.336999\pi$
$113$	10.4174			0.979989			0.489994	+	0.871726i	$0.336999\pi$
$114$	−2.83485			−0.265508
$115$		0			0
$116$			3.79129i			0.352012i
$117$	−0.791288			−0.0731546
$118$		−	1.58258i		−	0.145688i
$119$		−	15.1652i		−	1.39019i
$120$		0			0
$121$	−10.3739			−0.943079
$122$			12.7913i			1.15807i
$123$		−	17.5390i		−	1.58144i
$124$			8.37386i			0.751995i
$125$		0			0
$126$		−	0.417424i		−	0.0371871i
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	−1.00000			−0.0883883
$129$		−	10.7477i		−	0.946285i
$130$		0			0
$131$		−	16.7477i		−	1.46326i	−0.681704	−	0.731628i	$-0.738761\pi$
$131$		−	16.7477i		−	1.46326i	0.681704	−	0.731628i	$-0.261239\pi$
$132$			1.41742i			0.123371i
$133$	3.16515			0.274453
$134$		−	6.37386i		−	0.550618i
$135$		0			0
$136$	−7.58258			−0.650201
$137$		−	3.62614i		−	0.309802i	−0.987930	−	0.154901i	$-0.950494\pi$
$137$		−	3.62614i		−	0.309802i	0.987930	−	0.154901i	$-0.0495058\pi$
$138$			6.79129i			0.578113i
$139$	13.3739			1.13436			0.567178	−	0.823595i	$-0.308035\pi$
$139$	13.3739			1.13436			0.567178	+	0.823595i	$0.308035\pi$
$140$		0			0
$141$	−13.5826			−1.14386
$142$	9.16515			0.769122
$143$	−3.00000			−0.250873
$144$	−0.208712			−0.0173927
$145$		0			0
$146$		−	4.37386i		−	0.361984i
$147$		−	5.37386i		−	0.443229i
$148$	−4.58258	+	4.00000i	−0.376685	+	0.328798i
$149$	4.41742			0.361889			0.180945	−	0.983493i	$-0.442084\pi$
$149$	4.41742			0.361889			0.180945	+	0.983493i	$0.442084\pi$
$150$		0			0
$151$	−14.7477			−1.20015			−0.600077	−	0.799943i	$-0.704863\pi$
$151$	−14.7477			−1.20015			−0.600077	+	0.799943i	$0.704863\pi$
$152$		−	1.58258i		−	0.128364i
$153$	−1.58258			−0.127944
$154$		−	1.58258i		−	0.127528i
$155$		0			0
$156$		−	6.79129i		−	0.543738i
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$		−	8.20871i		−	0.653050i
$159$	2.83485			0.224818
$160$		0			0
$161$		−	7.58258i		−	0.597591i
$162$	9.58258			0.752878
$163$	−19.5826			−1.53383			−0.766913	−	0.641751i	$-0.778208\pi$
$163$	−19.5826			−1.53383			−0.766913	+	0.641751i	$0.778208\pi$
$164$	9.79129			0.764571
$165$		0			0
$166$			15.1652i			1.17704i
$167$	−21.9564			−1.69904			−0.849520	−	0.527556i	$-0.823108\pi$
$167$	−21.9564			−1.69904			−0.849520	+	0.527556i	$0.823108\pi$
$168$	3.58258			0.276402
$169$	1.37386			0.105682
$170$		0			0
$171$		−	0.330303i		−	0.0252589i
$172$	6.00000			0.457496
$173$		−	15.1652i		−	1.15299i	−0.817102	−	0.576493i	$-0.804421\pi$
$173$		−	15.1652i		−	1.15299i	0.817102	−	0.576493i	$-0.195579\pi$
$174$	−6.79129			−0.514846
$175$		0			0
$176$	−0.791288			−0.0596456
$177$	2.83485			0.213080
$178$		−	6.00000i		−	0.449719i
$179$			1.58258i			0.118287i	0.998249	+	0.0591436i	$0.0188370\pi$
$179$			1.58258i			0.118287i	−0.998249	+	0.0591436i	$0.981163\pi$
$180$		0			0
$181$	8.74773			0.650213			0.325107	−	0.945677i	$-0.394600\pi$
$181$	8.74773			0.650213			0.325107	+	0.945677i	$0.394600\pi$
$182$			7.58258i			0.562058i
$183$	−22.9129			−1.69377
$184$	−3.79129			−0.279497
$185$		0			0
$186$	−15.0000			−1.09985
$187$	−6.00000			−0.438763
$188$		−	7.58258i		−	0.553016i
$189$	−10.0000			−0.727393
$190$		0			0
$191$		−	8.37386i		−	0.605912i	−0.953005	−	0.302956i	$-0.902027\pi$
$191$		−	8.37386i		−	0.605912i	0.953005	−	0.302956i	$-0.0979734\pi$
$192$		−	1.79129i		−	0.129275i
$193$	18.3303			1.31944			0.659722	−	0.751510i	$-0.270674\pi$
$193$	18.3303			1.31944			0.659722	+	0.751510i	$0.270674\pi$
$194$	13.5826			0.975172
$195$		0			0
$196$	3.00000			0.214286
$197$		−	19.9129i		−	1.41873i	−0.704839	−	0.709367i	$-0.748981\pi$
$197$		−	19.9129i		−	1.41873i	0.704839	−	0.709367i	$-0.251019\pi$
$198$	−0.165151			−0.0117368
$199$		−	15.1652i		−	1.07503i	−0.843254	−	0.537515i	$-0.819363\pi$
$199$		−	15.1652i		−	1.07503i	0.843254	−	0.537515i	$-0.180637\pi$
$200$		0			0
$201$	11.4174			0.805323
$202$	−7.58258			−0.533508
$203$	7.58258			0.532192
$204$		−	13.5826i		−	0.950971i
$205$		0			0
$206$	6.79129			0.473171
$207$	−0.791288			−0.0549983
$208$	3.79129			0.262879
$209$		−	1.25227i		−	0.0866215i
$210$		0			0
$211$	10.3739			0.714166			0.357083	−	0.934073i	$-0.383771\pi$
$211$	10.3739			0.714166			0.357083	+	0.934073i	$0.383771\pi$
$212$			1.58258i			0.108692i
$213$			16.4174i			1.12490i
$214$			5.37386i			0.367350i
$215$		0			0
$216$			5.00000i			0.340207i
$217$	16.7477			1.13691
$218$		−	6.00000i		−	0.406371i
$219$	7.83485			0.529430
$220$		0			0
$221$	28.7477			1.93378
$222$	−7.16515	−	8.20871i	−0.480893	−	0.550933i
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$			2.00000i			0.133631i
$225$		0			0
$226$	−10.4174			−0.692957
$227$	−25.9129			−1.71990			−0.859949	−	0.510380i	$-0.829505\pi$
$227$	−25.9129			−1.71990			−0.859949	+	0.510380i	$0.829505\pi$
$228$	2.83485			0.187742
$229$	−6.74773			−0.445902			−0.222951	−	0.974830i	$-0.571569\pi$
$229$	−6.74773			−0.445902			−0.222951	+	0.974830i	$0.571569\pi$
$230$		0			0
$231$	2.83485			0.186519
$232$		−	3.79129i		−	0.248910i
$233$			16.1216i			1.05616i	0.849194	+	0.528080i	$0.177088\pi$
$233$			16.1216i			1.05616i	−0.849194	+	0.528080i	$0.822912\pi$
$234$	0.791288			0.0517281
$235$		0			0
$236$			1.58258i			0.103017i
$237$	14.7042			0.955138
$238$			15.1652i			0.983011i
$239$			2.04356i			0.132187i	0.997813	+	0.0660935i	$0.0210536\pi$
$239$			2.04356i			0.132187i	−0.997813	+	0.0660935i	$0.978946\pi$
$240$		0			0
$241$		−	4.41742i		−	0.284551i	−0.989827	−	0.142276i	$-0.954558\pi$
$241$		−	4.41742i		−	0.284551i	0.989827	−	0.142276i	$-0.0454419\pi$
$242$	10.3739			0.666857
$243$			2.16515i			0.138895i
$244$		−	12.7913i		−	0.818878i
$245$		0			0
$246$			17.5390i			1.11825i
$247$			6.00000i			0.381771i
$248$		−	8.37386i		−	0.531741i
$249$	−27.1652			−1.72152
$250$		0			0
$251$		−	4.41742i		−	0.278825i	−0.990234	−	0.139413i	$-0.955479\pi$
$251$		−	4.41742i		−	0.278825i	0.990234	−	0.139413i	$-0.0445215\pi$
$252$			0.417424i			0.0262953i
$253$	−3.00000			−0.188608
$254$		−	8.00000i		−	0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−4.74773			−0.296155			−0.148078	−	0.988976i	$-0.547309\pi$
$257$	−4.74773			−0.296155			−0.148078	+	0.988976i	$0.547309\pi$
$258$			10.7477i			0.669124i
$259$	8.00000	+	9.16515i	0.497096	+	0.569495i
$260$		0			0
$261$		−	0.791288i		−	0.0489795i
$262$			16.7477i			1.03468i
$263$			27.1652i			1.67507i	0.546380	+	0.837537i	$0.316006\pi$
$263$			27.1652i			1.67507i	−0.546380	+	0.837537i	$0.683994\pi$
$264$		−	1.41742i		−	0.0872364i
$265$		0			0
$266$	−3.16515			−0.194068
$267$	10.7477			0.657750
$268$			6.37386i			0.389346i
$269$	−16.7477			−1.02113			−0.510563	−	0.859840i	$-0.670563\pi$
$269$	−16.7477			−1.02113			−0.510563	+	0.859840i	$0.670563\pi$
$270$		0			0
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$	7.58258			0.459761
$273$	−13.5826			−0.822055
$274$			3.62614i			0.219063i
$275$		0			0
$276$		−	6.79129i		−	0.408787i
$277$	−9.62614			−0.578378			−0.289189	−	0.957272i	$-0.593386\pi$
$277$	−9.62614			−0.578378			−0.289189	+	0.957272i	$0.593386\pi$
$278$	−13.3739			−0.802111
$279$		−	1.74773i		−	0.104634i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	13.5826			0.808831
$283$	−24.0000			−1.42665			−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000			−1.42665			−0.713326	+	0.700832i	$0.752812\pi$
$284$	−9.16515			−0.543852
$285$		0			0
$286$	3.00000			0.177394
$287$		−	19.5826i		−	1.15592i
$288$	0.208712			0.0122985
$289$	40.4955			2.38209
$290$		0			0
$291$			24.3303i			1.42627i
$292$			4.37386i			0.255961i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$			5.37386i			0.313410i
$295$		0			0
$296$	4.58258	−	4.00000i	0.266357	−	0.232495i
$297$			3.95644i			0.229576i
$298$	−4.41742			−0.255895
$299$	14.3739			0.831262
$300$		0			0
$301$		−	12.0000i		−	0.691669i
$302$	14.7477			0.848636
$303$		−	13.5826i		−	0.780299i
$304$			1.58258i			0.0907669i
$305$		0			0
$306$	1.58258			0.0904698
$307$			1.37386i			0.0784105i	0.999231	+	0.0392053i	$0.0124826\pi$
$307$			1.37386i			0.0784105i	−0.999231	+	0.0392053i	$0.987517\pi$
$308$			1.58258i			0.0901756i
$309$			12.1652i			0.692051i
$310$		0			0
$311$			23.3739i			1.32541i	0.748880	+	0.662705i	$0.230592\pi$
$311$			23.3739i			1.32541i	−0.748880	+	0.662705i	$0.769408\pi$
$312$			6.79129i			0.384481i
$313$	14.8348			0.838515			0.419258	−	0.907867i	$-0.362290\pi$
$313$	14.8348			0.838515			0.419258	+	0.907867i	$0.362290\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$			8.20871i			0.461776i
$317$		−	28.7477i		−	1.61463i	−0.590119	−	0.807317i	$-0.700919\pi$
$317$		−	28.7477i		−	1.61463i	0.590119	−	0.807317i	$-0.299081\pi$
$318$	−2.83485			−0.158970
$319$		−	3.00000i		−	0.167968i
$320$		0			0
$321$	−9.62614			−0.537279
$322$			7.58258i			0.422560i
$323$			12.0000i			0.667698i
$324$	−9.58258			−0.532365
$325$		0			0
$326$	19.5826			1.08458
$327$	10.7477			0.594351
$328$	−9.79129			−0.540633
$329$	−15.1652			−0.836082
$330$		0			0
$331$			25.5826i			1.40615i	0.711118	+	0.703073i	$0.248189\pi$
$331$			25.5826i			1.40615i	−0.711118	+	0.703073i	$0.751811\pi$
$332$		−	15.1652i		−	0.832296i
$333$	0.956439	−	0.834849i	0.0524125	−	0.0457494i
$334$	21.9564			1.20140
$335$		0			0
$336$	−3.58258			−0.195446
$337$			33.1216i			1.80425i	0.431477	+	0.902124i	$0.357993\pi$
$337$			33.1216i			1.80425i	−0.431477	+	0.902124i	$0.642007\pi$
$338$	−1.37386			−0.0747283
$339$		−	18.6606i		−	1.01350i
$340$		0			0
$341$		−	6.62614i		−	0.358825i
$342$			0.330303i			0.0178607i
$343$		−	20.0000i		−	1.07990i
$344$	−6.00000			−0.323498
$345$		0			0
$346$			15.1652i			0.815284i
$347$	7.58258			0.407054			0.203527	−	0.979069i	$-0.434760\pi$
$347$	7.58258			0.407054			0.203527	+	0.979069i	$0.434760\pi$
$348$	6.79129			0.364051
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		−	18.9564i		−	1.01182i
$352$	0.791288			0.0421758
$353$	1.58258			0.0842320			0.0421160	−	0.999113i	$-0.486590\pi$
$353$	1.58258			0.0842320			0.0421160	+	0.999113i	$0.486590\pi$
$354$	−2.83485			−0.150671
$355$		0			0
$356$			6.00000i			0.317999i
$357$	−27.1652			−1.43773
$358$		−	1.58258i		−	0.0836417i
$359$	8.83485			0.466285			0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485			0.466285			0.233143	+	0.972443i	$0.425099\pi$
$360$		0			0
$361$	16.4955			0.868182
$362$	−8.74773			−0.459770
$363$			18.5826i			0.975332i
$364$		−	7.58258i		−	0.397435i
$365$		0			0
$366$	22.9129			1.19768
$367$		−	5.25227i		−	0.274166i	−0.990560	−	0.137083i	$-0.956227\pi$
$367$		−	5.25227i		−	0.274166i	0.990560	−	0.137083i	$-0.0437728\pi$
$368$	3.79129			0.197635
$369$	−2.04356			−0.106384
$370$		0			0
$371$	3.16515			0.164326
$372$	15.0000			0.777714
$373$			12.7477i			0.660052i	0.943972	+	0.330026i	$0.107058\pi$
$373$			12.7477i			0.660052i	−0.943972	+	0.330026i	$0.892942\pi$
$374$	6.00000			0.310253
$375$		0			0
$376$			7.58258i			0.391041i
$377$			14.3739i			0.740292i
$378$	10.0000			0.514344
$379$	−20.1216			−1.03358			−0.516788	−	0.856113i	$-0.672873\pi$
$379$	−20.1216			−1.03358			−0.516788	+	0.856113i	$0.672873\pi$
$380$		0			0
$381$	14.3303			0.734164
$382$			8.37386i			0.428444i
$383$	18.3303			0.936635			0.468317	−	0.883560i	$-0.344860\pi$
$383$	18.3303			0.936635			0.468317	+	0.883560i	$0.344860\pi$
$384$			1.79129i			0.0914113i
$385$		0			0
$386$	−18.3303			−0.932988
$387$	−1.25227			−0.0636566
$388$	−13.5826			−0.689551
$389$			32.8693i			1.66654i	0.552866	+	0.833270i	$0.313534\pi$
$389$			32.8693i			1.66654i	−0.552866	+	0.833270i	$0.686466\pi$
$390$		0			0
$391$	28.7477			1.45384
$392$	−3.00000			−0.151523
$393$	−30.0000			−1.51330
$394$			19.9129i			1.00320i
$395$		0			0
$396$	0.165151			0.00829917
$397$			14.7477i			0.740167i	0.928998	+	0.370084i	$0.120671\pi$
$397$			14.7477i			0.740167i	−0.928998	+	0.370084i	$0.879329\pi$
$398$			15.1652i			0.760160i
$399$		−	5.66970i		−	0.283840i
$400$		0			0
$401$			16.7477i			0.836342i	0.908368	+	0.418171i	$0.137329\pi$
$401$			16.7477i			0.836342i	−0.908368	+	0.418171i	$0.862671\pi$
$402$	−11.4174			−0.569449
$403$			31.7477i			1.58147i
$404$	7.58258			0.377247
$405$		0			0
$406$	−7.58258			−0.376317
$407$	3.62614	−	3.16515i	0.179741	−	0.156891i
$408$			13.5826i			0.672438i
$409$			27.1652i			1.34323i	0.740900	+	0.671615i	$0.234399\pi$
$409$			27.1652i			1.34323i	−0.740900	+	0.671615i	$0.765601\pi$
$410$		0			0
$411$	−6.49545			−0.320397
$412$	−6.79129			−0.334583
$413$	3.16515			0.155747
$414$	0.791288			0.0388897
$415$		0			0
$416$	−3.79129			−0.185883
$417$		−	23.9564i		−	1.17315i
$418$			1.25227i			0.0612507i
$419$	−2.20871			−0.107903			−0.0539513	−	0.998544i	$-0.517182\pi$
$419$	−2.20871			−0.107903			−0.0539513	+	0.998544i	$0.517182\pi$
$420$		0			0
$421$			18.9564i			0.923880i	0.886911	+	0.461940i	$0.152847\pi$
$421$			18.9564i			0.923880i	−0.886911	+	0.461940i	$0.847153\pi$
$422$	−10.3739			−0.504992
$423$			1.58258i			0.0769475i
$424$		−	1.58258i		−	0.0768567i
$425$		0			0
$426$		−	16.4174i		−	0.795427i
$427$	−25.5826			−1.23803
$428$		−	5.37386i		−	0.259755i
$429$			5.37386i			0.259453i
$430$		0			0
$431$			8.83485i			0.425560i	0.977100	+	0.212780i	$0.0682517\pi$
$431$			8.83485i			0.425560i	−0.977100	+	0.212780i	$0.931748\pi$
$432$		−	5.00000i		−	0.240563i
$433$		−	10.6261i		−	0.510660i	−0.966854	−	0.255330i	$-0.917816\pi$
$433$		−	10.6261i		−	0.510660i	0.966854	−	0.255330i	$-0.0821840\pi$
$434$	−16.7477			−0.803917
$435$		0			0
$436$			6.00000i			0.287348i
$437$			6.00000i			0.287019i
$438$	−7.83485			−0.374364
$439$			12.6261i			0.602613i	0.953527	+	0.301306i	$0.0974227\pi$
$439$			12.6261i			0.602613i	−0.953527	+	0.301306i	$0.902577\pi$
$440$		0			0
$441$	−0.626136			−0.0298160
$442$	−28.7477			−1.36739
$443$		−	27.9564i		−	1.32825i	−0.747621	−	0.664125i	$-0.768804\pi$
$443$		−	27.9564i		−	1.32825i	0.747621	−	0.664125i	$-0.231196\pi$
$444$	7.16515	+	8.20871i	0.340043	+	0.389568i
$445$		0			0
$446$			14.0000i			0.662919i
$447$		−	7.91288i		−	0.374266i
$448$		−	2.00000i		−	0.0944911i
$449$		−	2.83485i		−	0.133785i	−0.997760	−	0.0668924i	$-0.978692\pi$
$449$		−	2.83485i		−	0.133785i	0.997760	−	0.0668924i	$-0.0213084\pi$
$450$		0			0
$451$	−7.74773			−0.364826
$452$	10.4174			0.489994
$453$			26.4174i			1.24120i
$454$	25.9129			1.21615
$455$		0			0
$456$	−2.83485			−0.132754
$457$	−21.4955			−1.00551			−0.502757	−	0.864428i	$-0.667681\pi$
$457$	−21.4955			−1.00551			−0.502757	+	0.864428i	$0.667681\pi$
$458$	6.74773			0.315301
$459$		−	37.9129i		−	1.76962i
$460$		0			0
$461$		−	12.3303i		−	0.574279i	−0.957889	−	0.287140i	$-0.907296\pi$
$461$		−	12.3303i		−	0.574279i	0.957889	−	0.287140i	$-0.0927044\pi$
$462$	−2.83485			−0.131889
$463$	−29.7042			−1.38047			−0.690235	−	0.723585i	$-0.742493\pi$
$463$	−29.7042			−1.38047			−0.690235	+	0.723585i	$0.742493\pi$
$464$			3.79129i			0.176006i
$465$		0			0
$466$		−	16.1216i		−	0.746818i
$467$	16.4174			0.759708			0.379854	−	0.925046i	$-0.375974\pi$
$467$	16.4174			0.759708			0.379854	+	0.925046i	$0.375974\pi$
$468$	−0.791288			−0.0365773
$469$	12.7477			0.588635
$470$		0			0
$471$	−3.58258			−0.165076
$472$		−	1.58258i		−	0.0728440i
$473$	−4.74773			−0.218301
$474$	−14.7042			−0.675385
$475$		0			0
$476$		−	15.1652i		−	0.695094i
$477$		−	0.330303i		−	0.0151235i
$478$		−	2.04356i		−	0.0934703i
$479$			3.79129i			0.173228i	0.996242	+	0.0866142i	$0.0276047\pi$
$479$			3.79129i			0.173228i	−0.996242	+	0.0866142i	$0.972395\pi$
$480$		0			0
$481$	−17.3739	+	15.1652i	−0.792180	+	0.691471i
$482$			4.41742i			0.201208i
$483$	−13.5826			−0.618029
$484$	−10.3739			−0.471539
$485$		0			0
$486$		−	2.16515i		−	0.0982133i
$487$	36.6606			1.66125			0.830625	−	0.556832i	$-0.187983\pi$
$487$	36.6606			1.66125			0.830625	+	0.556832i	$0.187983\pi$
$488$			12.7913i			0.579034i
$489$			35.0780i			1.58628i
$490$		0			0
$491$	5.37386			0.242519			0.121260	−	0.992621i	$-0.461307\pi$
$491$	5.37386			0.242519			0.121260	+	0.992621i	$0.461307\pi$
$492$		−	17.5390i		−	0.790720i
$493$			28.7477i			1.29473i
$494$		−	6.00000i		−	0.269953i
$495$		0			0
$496$			8.37386i			0.375998i
$497$			18.3303i			0.822226i
$498$	27.1652			1.21730
$499$		−	10.7477i		−	0.481134i	−0.970632	−	0.240567i	$-0.922667\pi$
$499$		−	10.7477i		−	0.481134i	0.970632	−	0.240567i	$-0.0773334\pi$
$500$		0			0
$501$			39.3303i			1.75715i
$502$			4.41742i			0.197159i
$503$	−15.6261			−0.696735			−0.348367	−	0.937358i	$-0.613264\pi$
$503$	−15.6261			−0.696735			−0.348367	+	0.937358i	$0.613264\pi$
$504$		−	0.417424i		−	0.0185936i
$505$		0			0
$506$	3.00000			0.133366
$507$		−	2.46099i		−	0.109296i
$508$			8.00000i			0.354943i
$509$	−4.41742			−0.195799			−0.0978994	−	0.995196i	$-0.531212\pi$
$509$	−4.41742			−0.195799			−0.0978994	+	0.995196i	$0.531212\pi$
$510$		0			0
$511$	8.74773			0.386977
$512$	−1.00000			−0.0441942
$513$	7.91288			0.349362
$514$	4.74773			0.209413
$515$		0			0
$516$		−	10.7477i		−	0.473142i
$517$			6.00000i			0.263880i
$518$	−8.00000	−	9.16515i	−0.351500	−	0.402694i
$519$	−27.1652			−1.19242
$520$		0			0
$521$	−9.16515			−0.401533			−0.200766	−	0.979639i	$-0.564343\pi$
$521$	−9.16515			−0.401533			−0.200766	+	0.979639i	$0.564343\pi$
$522$			0.791288i			0.0346337i
$523$	−15.1652			−0.663126			−0.331563	−	0.943433i	$-0.607576\pi$
$523$	−15.1652			−0.663126			−0.331563	+	0.943433i	$0.607576\pi$
$524$		−	16.7477i		−	0.731628i
$525$		0			0
$526$		−	27.1652i		−	1.18446i
$527$			63.4955i			2.76591i
$528$			1.41742i			0.0616855i
$529$	−8.62614			−0.375049
$530$		0			0
$531$		−	0.330303i		−	0.0143339i
$532$	3.16515			0.137227
$533$	37.1216			1.60791
$534$	−10.7477			−0.465100
$535$		0			0
$536$		−	6.37386i		−	0.275309i
$537$	2.83485			0.122333
$538$	16.7477			0.722046
$539$	−2.37386			−0.102250
$540$		0			0
$541$			12.7913i			0.549940i	0.961453	+	0.274970i	$0.0886680\pi$
$541$			12.7913i			0.549940i	−0.961453	+	0.274970i	$0.911332\pi$
$542$	−22.0000			−0.944981
$543$		−	15.6697i		−	0.672451i
$544$	−7.58258			−0.325100
$545$		0			0
$546$	13.5826			0.581281
$547$	−25.9129			−1.10795			−0.553977	−	0.832532i	$-0.686891\pi$
$547$	−25.9129			−1.10795			−0.553977	+	0.832532i	$0.686891\pi$
$548$		−	3.62614i		−	0.154901i
$549$			2.66970i			0.113940i
$550$		0			0
$551$	−6.00000			−0.255609
$552$			6.79129i			0.289056i
$553$	16.4174			0.698140
$554$	9.62614			0.408975
$555$		0			0
$556$	13.3739			0.567178
$557$	−8.70417			−0.368807			−0.184404	−	0.982851i	$-0.559035\pi$
$557$	−8.70417			−0.368807			−0.184404	+	0.982851i	$0.559035\pi$
$558$			1.74773i			0.0739872i
$559$	22.7477			0.962126
$560$		0			0
$561$			10.7477i			0.453769i
$562$		0			0
$563$	−10.7477			−0.452963			−0.226481	−	0.974016i	$-0.572722\pi$
$563$	−10.7477			−0.452963			−0.226481	+	0.974016i	$0.572722\pi$
$564$	−13.5826			−0.571930
$565$		0			0
$566$	24.0000			1.00880
$567$			19.1652i			0.804861i
$568$	9.16515			0.384561
$569$		−	45.1652i		−	1.89342i	−0.322085	−	0.946711i	$-0.604384\pi$
$569$		−	45.1652i		−	1.89342i	0.322085	−	0.946711i	$-0.395616\pi$
$570$		0			0
$571$	−14.6261			−0.612085			−0.306042	−	0.952018i	$-0.599005\pi$
$571$	−14.6261			−0.612085			−0.306042	+	0.952018i	$0.599005\pi$
$572$	−3.00000			−0.125436
$573$	−15.0000			−0.626634
$574$			19.5826i			0.817361i
$575$		0			0
$576$	−0.208712			−0.00869634
$577$	−13.5826			−0.565450			−0.282725	−	0.959201i	$-0.591238\pi$
$577$	−13.5826			−0.565450			−0.282725	+	0.959201i	$0.591238\pi$
$578$	−40.4955			−1.68439
$579$		−	32.8348i		−	1.36457i
$580$		0			0
$581$	−30.3303			−1.25831
$582$		−	24.3303i		−	1.00852i
$583$		−	1.25227i		−	0.0518638i
$584$		−	4.37386i		−	0.180992i
$585$		0			0
$586$		−	6.00000i		−	0.247858i
$587$	19.9129			0.821892			0.410946	−	0.911660i	$-0.365198\pi$
$587$	19.9129			0.821892			0.410946	+	0.911660i	$0.365198\pi$
$588$		−	5.37386i		−	0.221614i
$589$	−13.2523			−0.546050
$590$		0			0
$591$	−35.6697			−1.46726
$592$	−4.58258	+	4.00000i	−0.188343	+	0.164399i
$593$			18.7913i			0.771666i	0.922569	+	0.385833i	$0.126086\pi$
$593$			18.7913i			0.771666i	−0.922569	+	0.385833i	$0.873914\pi$
$594$		−	3.95644i		−	0.162335i
$595$		0			0
$596$	4.41742			0.180945
$597$	−27.1652			−1.11180
$598$	−14.3739			−0.587791
$599$	−8.83485			−0.360982			−0.180491	−	0.983577i	$-0.557769\pi$
$599$	−8.83485			−0.360982			−0.180491	+	0.983577i	$0.557769\pi$
$600$		0			0
$601$	−33.1216			−1.35106			−0.675529	−	0.737333i	$-0.736085\pi$
$601$	−33.1216			−1.35106			−0.675529	+	0.737333i	$0.736085\pi$
$602$			12.0000i			0.489083i
$603$		−	1.33030i		−	0.0541741i
$604$	−14.7477			−0.600077
$605$		0			0
$606$			13.5826i			0.551754i
$607$	26.5390			1.07719			0.538593	−	0.842566i	$-0.318956\pi$
$607$	26.5390			1.07719			0.538593	+	0.842566i	$0.318956\pi$
$608$		−	1.58258i		−	0.0641819i
$609$		−	13.5826i		−	0.550394i
$610$		0			0
$611$		−	28.7477i		−	1.16301i
$612$	−1.58258			−0.0639718
$613$			49.4955i			1.99910i	0.0299539	+	0.999551i	$0.490464\pi$
$613$			49.4955i			1.99910i	−0.0299539	+	0.999551i	$0.509536\pi$
$614$		−	1.37386i		−	0.0554446i
$615$		0			0
$616$		−	1.58258i		−	0.0637638i
$617$		−	23.0436i		−	0.927699i	−0.885914	−	0.463849i	$-0.846468\pi$
$617$		−	23.0436i		−	0.927699i	0.885914	−	0.463849i	$-0.153532\pi$
$618$		−	12.1652i		−	0.489354i
$619$	35.1216			1.41166			0.705828	−	0.708383i	$-0.250575\pi$
$619$	35.1216			1.41166			0.705828	+	0.708383i	$0.250575\pi$
$620$		0			0
$621$		−	18.9564i		−	0.760696i
$622$		−	23.3739i		−	0.937207i
$623$	12.0000			0.480770
$624$		−	6.79129i		−	0.271869i
$625$		0			0
$626$	−14.8348			−0.592920
$627$	−2.24318			−0.0895840
$628$		−	2.00000i		−	0.0798087i
$629$	−34.7477	+	30.3303i	−1.38548	+	1.20935i
$630$		0			0
$631$		−	18.9564i		−	0.754644i	−0.926082	−	0.377322i	$-0.876845\pi$
$631$		−	18.9564i		−	0.754644i	0.926082	−	0.377322i	$-0.123155\pi$
$632$		−	8.20871i		−	0.326525i
$633$		−	18.5826i		−	0.738591i
$634$			28.7477i			1.14172i
$635$		0			0
$636$	2.83485			0.112409
$637$	11.3739			0.450649
$638$			3.00000i			0.118771i
$639$	1.91288			0.0756723
$640$		0			0
$641$	−3.46099			−0.136701			−0.0683503	−	0.997661i	$-0.521774\pi$
$641$	−3.46099			−0.136701			−0.0683503	+	0.997661i	$0.521774\pi$
$642$	9.62614			0.379913
$643$	39.4955			1.55755			0.778774	−	0.627304i	$-0.215842\pi$
$643$	39.4955			1.55755			0.778774	+	0.627304i	$0.215842\pi$
$644$		−	7.58258i		−	0.298795i
$645$		0			0
$646$		−	12.0000i		−	0.472134i
$647$	17.7042			0.696023			0.348011	−	0.937490i	$-0.386857\pi$
$647$	17.7042			0.696023			0.348011	+	0.937490i	$0.386857\pi$
$648$	9.58258			0.376439
$649$		−	1.25227i		−	0.0491560i
$650$		0			0
$651$		−	30.0000i		−	1.17579i
$652$	−19.5826			−0.766913
$653$	−0.626136			−0.0245026			−0.0122513	−	0.999925i	$-0.503900\pi$
$653$	−0.626136			−0.0245026			−0.0122513	+	0.999925i	$0.503900\pi$
$654$	−10.7477			−0.420269
$655$		0			0
$656$	9.79129			0.382286
$657$		−	0.912878i		−	0.0356148i
$658$	15.1652			0.591199
$659$	−11.0436			−0.430196			−0.215098	−	0.976592i	$-0.569007\pi$
$659$	−11.0436			−0.430196			−0.215098	+	0.976592i	$0.569007\pi$
$660$		0			0
$661$		−	32.2087i		−	1.25277i	−0.779512	−	0.626387i	$-0.784533\pi$
$661$		−	32.2087i		−	1.25277i	0.779512	−	0.626387i	$-0.215467\pi$
$662$		−	25.5826i		−	0.994295i
$663$		−	51.4955i		−	1.99992i
$664$			15.1652i			0.588522i
$665$		0			0
$666$	−0.956439	+	0.834849i	−0.0370613	+	0.0323497i
$667$			14.3739i			0.556558i
$668$	−21.9564			−0.849520
$669$	−25.0780			−0.969573
$670$		0			0
$671$			10.1216i			0.390740i
$672$	3.58258			0.138201
$673$			36.1216i			1.39238i	0.717855	+	0.696192i	$0.245124\pi$
$673$			36.1216i			1.39238i	−0.717855	+	0.696192i	$0.754876\pi$
$674$		−	33.1216i		−	1.27580i
$675$		0			0
$676$	1.37386			0.0528409
$677$			9.16515i			0.352245i	0.984368	+	0.176123i	$0.0563555\pi$
$677$			9.16515i			0.352245i	−0.984368	+	0.176123i	$0.943644\pi$
$678$			18.6606i			0.716656i
$679$			27.1652i			1.04250i
$680$		0			0
$681$			46.4174i			1.77872i
$682$			6.62614i			0.253728i
$683$	31.5826			1.20847			0.604237	−	0.796805i	$-0.293478\pi$
$683$	31.5826			1.20847			0.604237	+	0.796805i	$0.293478\pi$
$684$		−	0.330303i		−	0.0126294i
$685$		0			0
$686$			20.0000i			0.763604i
$687$			12.0871i			0.461152i
$688$	6.00000			0.228748
$689$			6.00000i			0.228582i
$690$		0			0
$691$	25.4955			0.969893			0.484946	−	0.874544i	$-0.338839\pi$
$691$	25.4955			0.969893			0.484946	+	0.874544i	$0.338839\pi$
$692$		−	15.1652i		−	0.576493i
$693$		−	0.330303i		−	0.0125472i
$694$	−7.58258			−0.287831
$695$		0			0
$696$	−6.79129			−0.257423
$697$	74.2432			2.81216
$698$	10.0000			0.378506
$699$	28.8784			1.09228
$700$		0			0
$701$		−	3.95644i		−	0.149433i	−0.997205	−	0.0747163i	$-0.976195\pi$
$701$		−	3.95644i		−	0.149433i	0.997205	−	0.0747163i	$-0.0238051\pi$
$702$			18.9564i			0.715465i
$703$	−6.33030	−	7.25227i	−0.238752	−	0.273525i
$704$	−0.791288			−0.0298228
$705$		0			0
$706$	−1.58258			−0.0595610
$707$		−	15.1652i		−	0.570344i
$708$	2.83485			0.106540
$709$			23.2087i			0.871621i	0.900038	+	0.435811i	$0.143538\pi$
$709$			23.2087i			0.871621i	−0.900038	+	0.435811i	$0.856462\pi$
$710$		0			0
$711$		−	1.71326i		−	0.0642522i
$712$		−	6.00000i		−	0.224860i
$713$			31.7477i			1.18896i
$714$	27.1652			1.01663
$715$		0			0
$716$			1.58258i			0.0591436i
$717$	3.66061			0.136708
$718$	−8.83485			−0.329714
$719$	21.1652			0.789327			0.394663	−	0.918826i	$-0.370861\pi$
$719$	21.1652			0.789327			0.394663	+	0.918826i	$0.370861\pi$
$720$		0			0
$721$			13.5826i			0.505842i
$722$	−16.4955			−0.613897
$723$	−7.91288			−0.294283
$724$	8.74773			0.325107
$725$		0			0
$726$		−	18.5826i		−	0.689664i
$727$	−13.1216			−0.486653			−0.243326	−	0.969944i	$-0.578239\pi$
$727$	−13.1216			−0.486653			−0.243326	+	0.969944i	$0.578239\pi$
$728$			7.58258i			0.281029i
$729$	−24.8693			−0.921086
$730$		0			0
$731$	45.4955			1.68271
$732$	−22.9129			−0.846884
$733$		−	47.4955i		−	1.75428i	−0.480231	−	0.877142i	$-0.659447\pi$
$733$		−	47.4955i		−	1.75428i	0.480231	−	0.877142i	$-0.340553\pi$
$734$			5.25227i			0.193865i
$735$		0			0
$736$	−3.79129			−0.139749
$737$		−	5.04356i		−	0.185782i
$738$	2.04356			0.0752245
$739$	35.1216			1.29197			0.645984	−	0.763351i	$-0.276447\pi$
$739$	35.1216			1.29197			0.645984	+	0.763351i	$0.276447\pi$
$740$		0			0
$741$	10.7477			0.394828
$742$	−3.16515			−0.116196
$743$		−	15.1652i		−	0.556355i	−0.960530	−	0.278178i	$-0.910270\pi$
$743$		−	15.1652i		−	0.556355i	0.960530	−	0.278178i	$-0.0897304\pi$
$744$	−15.0000			−0.549927
$745$		0			0
$746$		−	12.7477i		−	0.466727i
$747$			3.16515i			0.115807i
$748$	−6.00000			−0.219382
$749$	−10.7477			−0.392713
$750$		0			0
$751$	−41.4955			−1.51419			−0.757095	−	0.653304i	$-0.773382\pi$
$751$	−41.4955			−1.51419			−0.757095	+	0.653304i	$0.773382\pi$
$752$		−	7.58258i		−	0.276508i
$753$	−7.91288			−0.288361
$754$		−	14.3739i		−	0.523465i
$755$		0			0
$756$	−10.0000			−0.363696
$757$	−35.2087			−1.27968			−0.639841	−	0.768507i	$-0.721000\pi$
$757$	−35.2087			−1.27968			−0.639841	+	0.768507i	$0.721000\pi$
$758$	20.1216			0.730849
$759$			5.37386i			0.195059i
$760$		0			0
$761$	14.2087			0.515065			0.257533	−	0.966270i	$-0.417090\pi$
$761$	14.2087			0.515065			0.257533	+	0.966270i	$0.417090\pi$
$762$	−14.3303			−0.519132
$763$	12.0000			0.434429
$764$		−	8.37386i		−	0.302956i
$765$		0			0
$766$	−18.3303			−0.662301
$767$			6.00000i			0.216647i
$768$		−	1.79129i		−	0.0646375i
$769$		−	11.6697i		−	0.420820i	−0.977613	−	0.210410i	$-0.932520\pi$
$769$		−	11.6697i		−	0.420820i	0.977613	−	0.210410i	$-0.0674799\pi$
$770$		0			0
$771$			8.50455i			0.306284i
$772$	18.3303			0.659722
$773$		−	31.9129i		−	1.14783i	−0.818916	−	0.573913i	$-0.805425\pi$
$773$		−	31.9129i		−	1.14783i	0.818916	−	0.573913i	$-0.194575\pi$
$774$	1.25227			0.0450120
$775$		0			0
$776$	13.5826			0.487586
$777$	16.4174	−	14.3303i	0.588972	−	0.514097i
$778$		−	32.8693i		−	1.17842i
$779$			15.4955i			0.555182i
$780$		0			0
$781$	7.25227			0.259507
$782$	−28.7477			−1.02802
$783$	18.9564			0.677448
$784$	3.00000			0.107143
$785$		0			0
$786$	30.0000			1.07006
$787$			8.00000i			0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$			8.00000i			0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$		−	19.9129i		−	0.709367i
$789$	48.6606			1.73236
$790$		0			0
$791$		−	20.8348i		−	0.740802i
$792$	−0.165151			−0.00586840
$793$		−	48.4955i		−	1.72212i
$794$		−	14.7477i		−	0.523377i
$795$		0			0
$796$		−	15.1652i		−	0.537515i
$797$	35.3739			1.25301			0.626503	−	0.779419i	$-0.284486\pi$
$797$	35.3739			1.25301			0.626503	+	0.779419i	$0.284486\pi$
$798$			5.66970i			0.200705i
$799$		−	57.4955i		−	2.03404i
$800$		0			0
$801$		−	1.25227i		−	0.0442469i
$802$		−	16.7477i		−	0.591383i
$803$		−	3.46099i		−	0.122136i
$804$	11.4174			0.402662
$805$		0			0
$806$		−	31.7477i		−	1.11827i
$807$			30.0000i			1.05605i
$808$	−7.58258			−0.266754
$809$		−	53.0780i		−	1.86612i	−0.359715	−	0.933062i	$-0.617126\pi$
$809$		−	53.0780i		−	1.86612i	0.359715	−	0.933062i	$-0.382874\pi$
$810$		0			0
$811$	−19.8693			−0.697706			−0.348853	−	0.937177i	$-0.613429\pi$
$811$	−19.8693			−0.697706			−0.348853	+	0.937177i	$0.613429\pi$
$812$	7.58258			0.266096
$813$		−	39.4083i		−	1.38211i
$814$	−3.62614	+	3.16515i	−0.127096	+	0.110938i
$815$		0			0
$816$		−	13.5826i		−	0.475485i
$817$			9.49545i			0.332204i
$818$		−	27.1652i		−	0.949807i
$819$			1.58258i			0.0552997i
$820$		0			0
$821$	3.16515			0.110465			0.0552323	−	0.998474i	$-0.482410\pi$
$821$	3.16515			0.110465			0.0552323	+	0.998474i	$0.482410\pi$
$822$	6.49545			0.226555
$823$			16.0000i			0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$			16.0000i			0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	6.79129			0.236586
$825$		0			0
$826$	−3.16515			−0.110130
$827$	33.1652			1.15327			0.576633	−	0.817004i	$-0.304366\pi$
$827$	33.1652			1.15327			0.576633	+	0.817004i	$0.304366\pi$
$828$	−0.791288			−0.0274992
$829$			17.0436i			0.591947i	0.955196	+	0.295974i	$0.0956441\pi$
$829$			17.0436i			0.591947i	−0.955196	+	0.295974i	$0.904356\pi$
$830$		0			0
$831$			17.2432i			0.598159i
$832$	3.79129			0.131439
$833$	22.7477			0.788162
$834$			23.9564i			0.829544i
$835$		0			0
$836$		−	1.25227i		−	0.0433108i
$837$	41.8693			1.44722
$838$	2.20871			0.0762987
$839$	−3.49545			−0.120676			−0.0603382	−	0.998178i	$-0.519218\pi$
$839$	−3.49545			−0.120676			−0.0603382	+	0.998178i	$0.519218\pi$
$840$		0			0
$841$	14.6261			0.504350
$842$		−	18.9564i		−	0.653282i
$843$		0			0
$844$	10.3739			0.357083
$845$		0			0
$846$		−	1.58258i		−	0.0544101i
$847$			20.7477i			0.712900i
$848$			1.58258i			0.0543459i
$849$			42.9909i			1.47544i
$850$		0			0
$851$	−17.3739	+	15.1652i	−0.595568	+	0.519855i
$852$			16.4174i			0.562452i
$853$	−44.7042			−1.53064			−0.765321	−	0.643649i	$-0.777420\pi$
$853$	−44.7042			−1.53064			−0.765321	+	0.643649i	$0.777420\pi$
$854$	25.5826			0.875418
$855$		0			0
$856$			5.37386i			0.183675i
$857$	33.1652			1.13290			0.566450	−	0.824096i	$-0.308316\pi$
$857$	33.1652			1.13290			0.566450	+	0.824096i	$0.308316\pi$
$858$		−	5.37386i		−	0.183461i
$859$		−	18.6606i		−	0.636692i	−0.947975	−	0.318346i	$-0.896873\pi$
$859$		−	18.6606i		−	0.636692i	0.947975	−	0.318346i	$-0.103127\pi$
$860$		0			0
$861$	−35.0780			−1.19546
$862$		−	8.83485i		−	0.300916i
$863$			19.2523i			0.655355i	0.944790	+	0.327677i	$0.106266\pi$
$863$			19.2523i			0.655355i	−0.944790	+	0.327677i	$0.893734\pi$
$864$			5.00000i			0.170103i
$865$		0			0
$866$			10.6261i			0.361091i
$867$		−	72.5390i		−	2.46355i
$868$	16.7477			0.568455
$869$		−	6.49545i		−	0.220343i
$870$		0			0
$871$			24.1652i			0.818805i
$872$		−	6.00000i		−	0.203186i
$873$	2.83485			0.0959451
$874$		−	6.00000i		−	0.202953i
$875$		0			0
$876$	7.83485			0.264715
$877$			11.2523i			0.379962i	0.981788	+	0.189981i	$0.0608427\pi$
$877$			11.2523i			0.379962i	−0.981788	+	0.189981i	$0.939157\pi$
$878$		−	12.6261i		−	0.426111i
$879$	10.7477			0.362512
$880$		0			0
$881$	−13.1216			−0.442078			−0.221039	−	0.975265i	$-0.570945\pi$
$881$	−13.1216			−0.442078			−0.221039	+	0.975265i	$0.570945\pi$
$882$	0.626136			0.0210831
$883$	−31.9129			−1.07395			−0.536977	−	0.843597i	$-0.680434\pi$
$883$	−31.9129			−1.07395			−0.536977	+	0.843597i	$0.680434\pi$
$884$	28.7477			0.966891
$885$		0			0
$886$			27.9564i			0.939215i
$887$		−	57.8258i		−	1.94160i	−0.239893	−	0.970799i	$-0.577112\pi$
$887$		−	57.8258i		−	1.94160i	0.239893	−	0.970799i	$-0.422888\pi$
$888$	−7.16515	−	8.20871i	−0.240447	−	0.275466i
$889$	16.0000			0.536623
$890$		0			0
$891$	7.58258			0.254026
$892$		−	14.0000i		−	0.468755i
$893$	12.0000			0.401565
$894$			7.91288i			0.264646i
$895$		0			0
$896$			2.00000i			0.0668153i
$897$		−	25.7477i		−	0.859692i
$898$			2.83485i			0.0946001i
$899$	−31.7477			−1.05885
$900$		0			0
$901$			12.0000i			0.399778i
$902$	7.74773			0.257971
$903$	−21.4955			−0.715324
$904$	−10.4174			−0.346478
$905$		0			0
$906$		−	26.4174i		−	0.877660i
$907$	−30.3303			−1.00710			−0.503551	−	0.863966i	$-0.667973\pi$
$907$	−30.3303			−1.00710			−0.503551	+	0.863966i	$0.667973\pi$
$908$	−25.9129			−0.859949
$909$	−1.58258			−0.0524907
$910$		0			0
$911$		−	17.6697i		−	0.585423i	−0.956201	−	0.292712i	$-0.905442\pi$
$911$		−	17.6697i		−	0.585423i	0.956201	−	0.292712i	$-0.0945576\pi$
$912$	2.83485			0.0938712
$913$			12.0000i			0.397142i
$914$	21.4955			0.711006
$915$		0			0
$916$	−6.74773			−0.222951
$917$	−33.4955			−1.10612
$918$			37.9129i			1.25131i
$919$			36.0000i			1.18753i	0.804638	+	0.593765i	$0.202359\pi$
$919$			36.0000i			1.18753i	−0.804638	+	0.593765i	$0.797641\pi$
$920$		0			0
$921$	2.46099			0.0810922
$922$			12.3303i			0.406077i
$923$	−34.7477			−1.14374
$924$	2.83485			0.0932597
$925$		0			0
$926$	29.7042			0.976139
$927$	1.41742			0.0465543
$928$		−	3.79129i		−	0.124455i
$929$	−8.37386			−0.274738			−0.137369	−	0.990520i	$-0.543865\pi$
$929$	−8.37386			−0.274738			−0.137369	+	0.990520i	$0.543865\pi$
$930$		0			0
$931$			4.74773i			0.155600i
$932$			16.1216i			0.528080i
$933$	41.8693			1.37074
$934$	−16.4174			−0.537195
$935$		0			0
$936$	0.791288			0.0258641
$937$			34.8693i			1.13913i	0.821946	+	0.569565i	$0.192889\pi$
$937$			34.8693i			1.13913i	−0.821946	+	0.569565i	$0.807111\pi$
$938$	−12.7477			−0.416228
$939$		−	26.5735i		−	0.867193i
$940$		0			0
$941$	−21.4955			−0.700732			−0.350366	−	0.936613i	$-0.613943\pi$
$941$	−21.4955			−0.700732			−0.350366	+	0.936613i	$0.613943\pi$
$942$	3.58258			0.116727
$943$	37.1216			1.20885
$944$			1.58258i			0.0515085i
$945$		0			0
$946$	4.74773			0.154362
$947$	−1.25227			−0.0406934			−0.0203467	−	0.999793i	$-0.506477\pi$
$947$	−1.25227			−0.0406934			−0.0203467	+	0.999793i	$0.506477\pi$
$948$	14.7042			0.477569
$949$			16.5826i			0.538293i
$950$		0			0
$951$	−51.4955			−1.66985
$952$			15.1652i			0.491505i
$953$		−	31.4519i		−	1.01883i	−0.860522	−	0.509413i	$-0.829862\pi$
$953$		−	31.4519i		−	1.01883i	0.860522	−	0.509413i	$-0.170138\pi$
$954$			0.330303i			0.0106939i
$955$		0			0
$956$			2.04356i			0.0660935i
$957$	−5.37386			−0.173712
$958$		−	3.79129i		−	0.122491i
$959$	−7.25227			−0.234188
$960$		0			0
$961$	−39.1216			−1.26199
$962$	17.3739	−	15.1652i	0.560156	−	0.488944i
$963$			1.12159i			0.0361428i
$964$		−	4.41742i		−	0.142276i
$965$		0			0
$966$	13.5826			0.437012
$967$	−21.9564			−0.706071			−0.353036	−	0.935610i	$-0.614851\pi$
$967$	−21.9564			−0.706071			−0.353036	+	0.935610i	$0.614851\pi$
$968$	10.3739			0.333429
$969$	21.4955			0.690533
$970$		0			0
$971$	50.3739			1.61657			0.808287	−	0.588789i	$-0.200395\pi$
$971$	50.3739			1.61657			0.808287	+	0.588789i	$0.200395\pi$
$972$			2.16515i			0.0694473i
$973$		−	26.7477i		−	0.857493i
$974$	−36.6606			−1.17468
$975$		0			0
$976$		−	12.7913i		−	0.409439i
$977$	41.0780			1.31420			0.657101	−	0.753802i	$-0.271782\pi$
$977$	41.0780			1.31420			0.657101	+	0.753802i	$0.271782\pi$
$978$		−	35.0780i		−	1.12167i
$979$		−	4.74773i		−	0.151738i
$980$		0			0
$981$		−	1.25227i		−	0.0399820i
$982$	−5.37386			−0.171487
$983$			18.3303i			0.584646i	0.956320	+	0.292323i	$0.0944282\pi$
$983$			18.3303i			0.584646i	−0.956320	+	0.292323i	$0.905572\pi$
$984$			17.5390i			0.559123i
$985$		0			0
$986$		−	28.7477i		−	0.915514i
$987$			27.1652i			0.864676i
$988$			6.00000i			0.190885i
$989$	22.7477			0.723336
$990$		0			0
$991$		−	18.9564i		−	0.602171i	−0.953597	−	0.301086i	$-0.902651\pi$
$991$		−	18.9564i		−	0.602171i	0.953597	−	0.301086i	$-0.0973490\pi$
$992$		−	8.37386i		−	0.265870i
$993$	45.8258			1.45424
$994$		−	18.3303i		−	0.581402i
$995$		0			0
$996$	−27.1652			−0.860761
$997$	−18.0000			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$998$			10.7477i			0.340213i
$999$	20.0000	+	22.9129i	0.632772	+	0.724931i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.c.g.1849.2		4
5.2	odd	4		1850.2.d.e.1701.1		4
5.3	odd	4		74.2.b.a.73.4	yes	4
5.4	even	2		1850.2.c.h.1849.3		4
15.8	even	4		666.2.c.b.73.2		4
20.3	even	4		592.2.g.c.369.1		4
37.36	even	2		1850.2.c.h.1849.2		4
40.3	even	4		2368.2.g.h.961.4		4
40.13	odd	4		2368.2.g.j.961.2		4
60.23	odd	4		5328.2.h.m.2737.3		4
185.43	even	4		2738.2.a.k.1.1		2
185.68	even	4		2738.2.a.h.1.1		2
185.73	odd	4		74.2.b.a.73.2	✓	4
185.147	odd	4		1850.2.d.e.1701.3		4
185.184	even	2	inner	1850.2.c.g.1849.3		4
555.443	even	4		666.2.c.b.73.3		4
740.443	even	4		592.2.g.c.369.2		4
1480.443	even	4		2368.2.g.h.961.3		4
1480.813	odd	4		2368.2.g.j.961.1		4
2220.443	odd	4		5328.2.h.m.2737.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.2	✓	4	185.73	odd	4
74.2.b.a.73.4	yes	4	5.3	odd	4
592.2.g.c.369.1		4	20.3	even	4
592.2.g.c.369.2		4	740.443	even	4
666.2.c.b.73.2		4	15.8	even	4
666.2.c.b.73.3		4	555.443	even	4
1850.2.c.g.1849.2		4	1.1	even	1	trivial
1850.2.c.g.1849.3		4	185.184	even	2	inner
1850.2.c.h.1849.2		4	37.36	even	2
1850.2.c.h.1849.3		4	5.4	even	2
1850.2.d.e.1701.1		4	5.2	odd	4
1850.2.d.e.1701.3		4	185.147	odd	4
2368.2.g.h.961.3		4	1480.443	even	4
2368.2.g.h.961.4		4	40.3	even	4
2368.2.g.j.961.1		4	1480.813	odd	4
2368.2.g.j.961.2		4	40.13	odd	4
2738.2.a.h.1.1		2	185.68	even	4
2738.2.a.k.1.1		2	185.43	even	4
5328.2.h.m.2737.2		4	2220.443	odd	4
5328.2.h.m.2737.3		4	60.23	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.79129i q^{3} +1.00000 q^{4} +1.79129i q^{6} -2.00000i q^{7} -1.00000 q^{8} -0.208712 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.79129i q^{3} +1.00000 q^{4} +1.79129i q^{6} -2.00000i q^{7} -1.00000 q^{8} -0.208712 q^{9} -0.791288 q^{11} -1.79129i q^{12} +3.79129 q^{13} +2.00000i q^{14} +1.00000 q^{16} +7.58258 q^{17} +0.208712 q^{18} +1.58258i q^{19} -3.58258 q^{21} +0.791288 q^{22} +3.79129 q^{23} +1.79129i q^{24} -3.79129 q^{26} -5.00000i q^{27} -2.00000i q^{28} +3.79129i q^{29} +8.37386i q^{31} -1.00000 q^{32} +1.41742i q^{33} -7.58258 q^{34} -0.208712 q^{36} +(-4.58258 + 4.00000i) q^{37} -1.58258i q^{38} -6.79129i q^{39} +9.79129 q^{41} +3.58258 q^{42} +6.00000 q^{43} -0.791288 q^{44} -3.79129 q^{46} -7.58258i q^{47} -1.79129i q^{48} +3.00000 q^{49} -13.5826i q^{51} +3.79129 q^{52} +1.58258i q^{53} +5.00000i q^{54} +2.00000i q^{56} +2.83485 q^{57} -3.79129i q^{58} +1.58258i q^{59} -12.7913i q^{61} -8.37386i q^{62} +0.417424i q^{63} +1.00000 q^{64} -1.41742i q^{66} +6.37386i q^{67} +7.58258 q^{68} -6.79129i q^{69} -9.16515 q^{71} +0.208712 q^{72} +4.37386i q^{73} +(4.58258 - 4.00000i) q^{74} +1.58258i q^{76} +1.58258i q^{77} +6.79129i q^{78} +8.20871i q^{79} -9.58258 q^{81} -9.79129 q^{82} -15.1652i q^{83} -3.58258 q^{84} -6.00000 q^{86} +6.79129 q^{87} +0.791288 q^{88} +6.00000i q^{89} -7.58258i q^{91} +3.79129 q^{92} +15.0000 q^{93} +7.58258i q^{94} +1.79129i q^{96} -13.5826 q^{97} -3.00000 q^{98} +0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 10 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 10 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 10 q^{9} + 6 q^{11} + 6 q^{13} + 4 q^{16} + 12 q^{17} + 10 q^{18} + 4 q^{21} - 6 q^{22} + 6 q^{23} - 6 q^{26} - 4 q^{32} - 12 q^{34} - 10 q^{36} + 30 q^{41} - 4 q^{42} + 24 q^{43} + 6 q^{44} - 6 q^{46} + 12 q^{49} + 6 q^{52} + 48 q^{57} + 4 q^{64} + 12 q^{68} + 10 q^{72} - 20 q^{81} - 30 q^{82} + 4 q^{84} - 24 q^{86} + 18 q^{87} - 6 q^{88} + 6 q^{92} + 60 q^{93} - 36 q^{97} - 12 q^{98} - 36 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 10 * q^9 + 6 * q^11 + 6 * q^13 + 4 * q^16 + 12 * q^17 + 10 * q^18 + 4 * q^21 - 6 * q^22 + 6 * q^23 - 6 * q^26 - 4 * q^32 - 12 * q^34 - 10 * q^36 + 30 * q^41 - 4 * q^42 + 24 * q^43 + 6 * q^44 - 6 * q^46 + 12 * q^49 + 6 * q^52 + 48 * q^57 + 4 * q^64 + 12 * q^68 + 10 * q^72 - 20 * q^81 - 30 * q^82 + 4 * q^84 - 24 * q^86 + 18 * q^87 - 6 * q^88 + 6 * q^92 + 60 * q^93 - 36 * q^97 - 12 * q^98 - 36 * q^99

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