LMFDB - Embedded newform 1850.2.d.i.1701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1701,1850)]

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([0, 1]))

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

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

chi := DirichletCharacter("1850.1701");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.d (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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + \cdots + 1 $ x^20 + 48x^18 + 878x^16 + 8102x^14 + 41081x^12 + 115688x^10 + 175041x^8 + 134990x^6 + 49782x^4 + 6932*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1701.1
Root			$-2.40359i$ of defining polynomial
Character	$\chi$	$=$	1850.1701
Dual form			1850.2.d.i.1701.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -3.40359 q^{3} -1.00000 q^{4} +3.40359i q^{6} +2.06225 q^{7} +1.00000i q^{8} +8.58443 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -3.40359 q^{3} -1.00000 q^{4} +3.40359i q^{6} +2.06225 q^{7} +1.00000i q^{8} +8.58443 q^{9} +3.77719 q^{11} +3.40359 q^{12} +2.88351i q^{13} -2.06225i q^{14} +1.00000 q^{16} +5.80724i q^{17} -8.58443i q^{18} +0.157282i q^{19} -7.01905 q^{21} -3.77719i q^{22} -5.41883i q^{23} -3.40359i q^{24} +2.88351 q^{26} -19.0071 q^{27} -2.06225 q^{28} +4.29061i q^{29} +0.425694i q^{31} -1.00000i q^{32} -12.8560 q^{33} +5.80724 q^{34} -8.58443 q^{36} +(-3.72698 - 4.80724i) q^{37} +0.157282 q^{38} -9.81429i q^{39} +0.923733 q^{41} +7.01905i q^{42} +10.8263i q^{43} -3.77719 q^{44} -5.41883 q^{46} +0.676445 q^{47} -3.40359 q^{48} -2.74713 q^{49} -19.7655i q^{51} -2.88351i q^{52} -9.87810 q^{53} +19.0071i q^{54} +2.06225i q^{56} -0.535322i q^{57} +4.29061 q^{58} +8.47192i q^{59} +1.23904i q^{61} +0.425694 q^{62} +17.7032 q^{63} -1.00000 q^{64} +12.8560i q^{66} +6.45516 q^{67} -5.80724i q^{68} +18.4435i q^{69} -3.28329 q^{71} +8.58443i q^{72} +0.980489 q^{73} +(-4.80724 + 3.72698i) q^{74} -0.157282i q^{76} +7.78950 q^{77} -9.81429 q^{78} -8.04725i q^{79} +38.9391 q^{81} -0.923733i q^{82} +11.9496 q^{83} +7.01905 q^{84} +10.8263 q^{86} -14.6035i q^{87} +3.77719i q^{88} +7.65857i q^{89} +5.94652i q^{91} +5.41883i q^{92} -1.44889i q^{93} -0.676445i q^{94} +3.40359i q^{96} +13.9571i q^{97} +2.74713i q^{98} +32.4250 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{4} + 16 q^{9}+O(q^{10}) $ 20 * q - 20 * q^4 + 16 * q^9	$ 20 q - 20 q^{4} + 16 q^{9} + 20 q^{16} - 24 q^{21} + 4 q^{26} + 36 q^{34} - 16 q^{36} - 8 q^{41} - 20 q^{46} + 16 q^{49} - 20 q^{64} - 40 q^{71} - 16 q^{74} + 116 q^{81} + 24 q^{84} + 20 q^{86} + 164 q^{99}+O(q^{100}) $ 20 * q - 20 * q^4 + 16 * q^9 + 20 * q^16 - 24 * q^21 + 4 * q^26 + 36 * q^34 - 16 * q^36 - 8 * q^41 - 20 * q^46 + 16 * q^49 - 20 * q^64 - 40 * q^71 - 16 * q^74 + 116 * q^81 + 24 * q^84 + 20 * q^86 + 164 * 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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$	−3.40359			−1.96506			−0.982532	−	0.186094i	$-0.940417\pi$
$3$	−3.40359			−1.96506			−0.982532	+	0.186094i	$0.940417\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			3.40359i			1.38951i
$7$	2.06225			0.779457			0.389728	−	0.920930i	$-0.372569\pi$
$7$	2.06225			0.779457			0.389728	+	0.920930i	$0.372569\pi$
$8$			1.00000i			0.353553i
$9$	8.58443			2.86148
$10$		0			0
$11$	3.77719			1.13886			0.569432	−	0.822038i	$-0.307163\pi$
$11$	3.77719			1.13886			0.569432	+	0.822038i	$0.307163\pi$
$12$	3.40359			0.982532
$13$			2.88351i			0.799742i	0.916571	+	0.399871i	$0.130945\pi$
$13$			2.88351i			0.799742i	−0.916571	+	0.399871i	$0.869055\pi$
$14$		−	2.06225i		−	0.551159i
$15$		0			0
$16$	1.00000			0.250000
$17$			5.80724i			1.40846i	0.709970	+	0.704232i	$0.248708\pi$
$17$			5.80724i			1.40846i	−0.709970	+	0.704232i	$0.751292\pi$
$18$		−	8.58443i		−	2.02337i
$19$			0.157282i			0.0360829i	0.999837	+	0.0180414i	$0.00574308\pi$
$19$			0.157282i			0.0360829i	−0.999837	+	0.0180414i	$0.994257\pi$
$20$		0			0
$21$	−7.01905			−1.53168
$22$		−	3.77719i		−	0.805299i
$23$		−	5.41883i		−	1.12990i	−0.825124	−	0.564952i	$-0.808895\pi$
$23$		−	5.41883i		−	1.12990i	0.825124	−	0.564952i	$-0.191105\pi$
$24$		−	3.40359i		−	0.694755i
$25$		0			0
$26$	2.88351			0.565503
$27$	−19.0071			−3.65792
$28$	−2.06225			−0.389728
$29$			4.29061i			0.796746i	0.917223	+	0.398373i	$0.130425\pi$
$29$			4.29061i			0.796746i	−0.917223	+	0.398373i	$0.869575\pi$
$30$		0			0
$31$			0.425694i			0.0764569i	0.999269	+	0.0382285i	$0.0121715\pi$
$31$			0.425694i			0.0764569i	−0.999269	+	0.0382285i	$0.987829\pi$
$32$		−	1.00000i		−	0.176777i
$33$	−12.8560			−2.23794
$34$	5.80724			0.995934
$35$		0			0
$36$	−8.58443			−1.43074
$37$	−3.72698	−	4.80724i	−0.612713	−	0.790306i
$37$	−3.72698	−	4.80724i	−0.612713	−	0.790306i
$38$	0.157282			0.0255144
$39$		−	9.81429i		−	1.57154i
$40$		0			0
$41$	0.923733			0.144263			0.0721314	−	0.997395i	$-0.477020\pi$
$41$	0.923733			0.144263			0.0721314	+	0.997395i	$0.477020\pi$
$42$			7.01905i			1.08306i
$43$			10.8263i			1.65099i	0.564406	+	0.825497i	$0.309105\pi$
$43$			10.8263i			1.65099i	−0.564406	+	0.825497i	$0.690895\pi$
$44$	−3.77719			−0.569432
$45$		0			0
$46$	−5.41883			−0.798963
$47$	0.676445			0.0986697			0.0493348	−	0.998782i	$-0.484290\pi$
$47$	0.676445			0.0986697			0.0493348	+	0.998782i	$0.484290\pi$
$48$	−3.40359			−0.491266
$49$	−2.74713			−0.392447
$50$		0			0
$51$		−	19.7655i		−	2.76772i
$52$		−	2.88351i		−	0.399871i
$53$	−9.87810			−1.35686			−0.678431	−	0.734664i	$-0.737340\pi$
$53$	−9.87810			−1.35686			−0.678431	+	0.734664i	$0.737340\pi$
$54$			19.0071i			2.58654i
$55$		0			0
$56$			2.06225i			0.275580i
$57$		−	0.535322i		−	0.0709052i
$58$	4.29061			0.563385
$59$			8.47192i			1.10295i	0.834192	+	0.551475i	$0.185935\pi$
$59$			8.47192i			1.10295i	−0.834192	+	0.551475i	$0.814065\pi$
$60$		0			0
$61$			1.23904i			0.158643i	0.996849	+	0.0793215i	$0.0252754\pi$
$61$			1.23904i			0.158643i	−0.996849	+	0.0793215i	$0.974725\pi$
$62$	0.425694			0.0540632
$63$	17.7032			2.23040
$64$	−1.00000			−0.125000
$65$		0			0
$66$			12.8560i			1.58246i
$67$	6.45516			0.788623			0.394312	−	0.918977i	$-0.370983\pi$
$67$	6.45516			0.788623			0.394312	+	0.918977i	$0.370983\pi$
$68$		−	5.80724i		−	0.704232i
$69$			18.4435i			2.22033i
$70$		0			0
$71$	−3.28329			−0.389655			−0.194828	−	0.980838i	$-0.562415\pi$
$71$	−3.28329			−0.389655			−0.194828	+	0.980838i	$0.562415\pi$
$72$			8.58443i			1.01168i
$73$	0.980489			0.114758			0.0573788	−	0.998352i	$-0.481726\pi$
$73$	0.980489			0.114758			0.0573788	+	0.998352i	$0.481726\pi$
$74$	−4.80724	+	3.72698i	−0.558831	+	0.433253i
$75$		0			0
$76$		−	0.157282i		−	0.0180414i
$77$	7.78950			0.887696
$78$	−9.81429			−1.11125
$79$		−	8.04725i		−	0.905387i	−0.891666	−	0.452693i	$-0.850463\pi$
$79$		−	8.04725i		−	0.905387i	0.891666	−	0.452693i	$-0.149537\pi$
$80$		0			0
$81$	38.9391			4.32657
$82$		−	0.923733i		−	0.102009i
$83$	11.9496			1.31164			0.655821	−	0.754916i	$-0.272323\pi$
$83$	11.9496			1.31164			0.655821	+	0.754916i	$0.272323\pi$
$84$	7.01905			0.765841
$85$		0			0
$86$	10.8263			1.16743
$87$		−	14.6035i		−	1.56566i
$88$			3.77719i			0.402649i
$89$			7.65857i			0.811807i	0.913916	+	0.405903i	$0.133043\pi$
$89$			7.65857i			0.811807i	−0.913916	+	0.405903i	$0.866957\pi$
$90$		0			0
$91$			5.94652i			0.623364i
$92$			5.41883i			0.564952i
$93$		−	1.44889i		−	0.150243i
$94$		−	0.676445i		−	0.0697700i
$95$		0			0
$96$			3.40359i			0.347378i
$97$			13.9571i			1.41712i	0.705649	+	0.708562i	$0.250656\pi$
$97$			13.9571i			1.41712i	−0.705649	+	0.708562i	$0.749344\pi$
$98$			2.74713i			0.277502i
$99$	32.4250			3.25883
$100$		0			0
$101$	−8.38279			−0.834119			−0.417059	−	0.908879i	$-0.636939\pi$
$101$	−8.38279			−0.834119			−0.417059	+	0.908879i	$0.636939\pi$
$102$	−19.7655			−1.95707
$103$		−	13.4727i		−	1.32751i	−0.747951	−	0.663753i	$-0.768962\pi$
$103$		−	13.4727i		−	1.32751i	0.747951	−	0.663753i	$-0.231038\pi$
$104$	−2.88351			−0.282751
$105$		0			0
$106$			9.87810i			0.959446i
$107$	−1.89614			−0.183306			−0.0916532	−	0.995791i	$-0.529215\pi$
$107$	−1.89614			−0.183306			−0.0916532	+	0.995791i	$0.529215\pi$
$108$	19.0071			1.82896
$109$			17.2044i			1.64789i	0.566672	+	0.823943i	$0.308231\pi$
$109$			17.2044i			1.64789i	−0.566672	+	0.823943i	$0.691769\pi$
$110$		0			0
$111$	12.6851	+	16.3619i	1.20402	+	1.55300i
$112$	2.06225			0.194864
$113$			1.17098i			0.110157i	0.998482	+	0.0550783i	$0.0175408\pi$
$113$			1.17098i			0.110157i	−0.998482	+	0.0550783i	$0.982459\pi$
$114$	−0.535322			−0.0501375
$115$		0			0
$116$		−	4.29061i		−	0.398373i
$117$			24.7533i			2.28844i
$118$	8.47192			0.779903
$119$			11.9760i			1.09784i
$120$		0			0
$121$	3.26714			0.297012
$122$	1.23904			0.112178
$123$	−3.14401			−0.283486
$124$		−	0.425694i		−	0.0382285i
$125$		0			0
$126$		−	17.7032i		−	1.57713i
$127$	16.1128			1.42978			0.714891	−	0.699236i	$-0.246476\pi$
$127$	16.1128			1.42978			0.714891	+	0.699236i	$0.246476\pi$
$128$			1.00000i			0.0883883i
$129$		−	36.8483i		−	3.24431i
$130$		0			0
$131$			9.64715i			0.842875i	0.906857	+	0.421438i	$0.138474\pi$
$131$			9.64715i			0.842875i	−0.906857	+	0.421438i	$0.861526\pi$
$132$	12.8560			1.11897
$133$			0.324354i			0.0281250i
$134$		−	6.45516i		−	0.557641i
$135$		0			0
$136$	−5.80724			−0.497967
$137$	−2.53592			−0.216658			−0.108329	−	0.994115i	$-0.534550\pi$
$137$	−2.53592			−0.216658			−0.108329	+	0.994115i	$0.534550\pi$
$138$	18.4435			1.57001
$139$	−7.56000			−0.641230			−0.320615	−	0.947210i	$-0.603890\pi$
$139$	−7.56000			−0.641230			−0.320615	+	0.947210i	$0.603890\pi$
$140$		0			0
$141$	−2.30234			−0.193892
$142$			3.28329i			0.275528i
$143$			10.8916i			0.910798i
$144$	8.58443			0.715369
$145$		0			0
$146$		−	0.980489i		−	0.0811458i
$147$	9.35011			0.771184
$148$	3.72698	+	4.80724i	0.306356	+	0.395153i
$149$	−6.72596			−0.551012			−0.275506	−	0.961299i	$-0.588845\pi$
$149$	−6.72596			−0.551012			−0.275506	+	0.961299i	$0.588845\pi$
$150$		0			0
$151$	−15.8051			−1.28620			−0.643101	−	0.765781i	$-0.722353\pi$
$151$	−15.8051			−1.28620			−0.643101	+	0.765781i	$0.722353\pi$
$152$	−0.157282			−0.0127572
$153$			49.8519i			4.03028i
$154$		−	7.78950i		−	0.627696i
$155$		0			0
$156$			9.81429i			0.785772i
$157$	8.50755			0.678976			0.339488	−	0.940610i	$-0.389746\pi$
$157$	8.50755			0.678976			0.339488	+	0.940610i	$0.389746\pi$
$158$	−8.04725			−0.640205
$159$	33.6210			2.66632
$160$		0			0
$161$		−	11.1750i		−	0.880712i
$162$		−	38.9391i		−	3.05935i
$163$			5.41321i			0.423995i	0.977270	+	0.211998i	$0.0679969\pi$
$163$			5.41321i			0.423995i	−0.977270	+	0.211998i	$0.932003\pi$
$164$	−0.923733			−0.0721314
$165$		0			0
$166$		−	11.9496i		−	0.927471i
$167$			2.08165i			0.161083i	0.996751	+	0.0805415i	$0.0256649\pi$
$167$			2.08165i			0.161083i	−0.996751	+	0.0805415i	$0.974335\pi$
$168$		−	7.01905i		−	0.541532i
$169$	4.68537			0.360413
$170$		0			0
$171$			1.35017i			0.103250i
$172$		−	10.8263i		−	0.825497i
$173$	−12.3562			−0.939423			−0.469712	−	0.882820i	$-0.655642\pi$
$173$	−12.3562			−0.939423			−0.469712	+	0.882820i	$0.655642\pi$
$174$	−14.6035			−1.10709
$175$		0			0
$176$	3.77719			0.284716
$177$		−	28.8349i		−	2.16737i
$178$	7.65857			0.574034
$179$		−	13.4571i		−	1.00583i	−0.864336	−	0.502915i	$-0.832261\pi$
$179$		−	13.4571i		−	1.00583i	0.864336	−	0.502915i	$-0.167739\pi$
$180$		0			0
$181$	−3.66608			−0.272498			−0.136249	−	0.990675i	$-0.543505\pi$
$181$	−3.66608			−0.272498			−0.136249	+	0.990675i	$0.543505\pi$
$182$	5.94652			0.440785
$183$		−	4.21719i		−	0.311744i
$184$	5.41883			0.399482
$185$		0			0
$186$	−1.44889			−0.106238
$187$			21.9350i			1.60405i
$188$	−0.676445			−0.0493348
$189$	−39.1974			−2.85119
$190$		0			0
$191$		−	18.5540i		−	1.34252i	−0.741221	−	0.671261i	$-0.765753\pi$
$191$		−	18.5540i		−	1.34252i	0.741221	−	0.671261i	$-0.234247\pi$
$192$	3.40359			0.245633
$193$		−	4.89778i		−	0.352550i	−0.984341	−	0.176275i	$-0.943595\pi$
$193$		−	4.89778i		−	0.352550i	0.984341	−	0.176275i	$-0.0564048\pi$
$194$	13.9571			1.00206
$195$		0			0
$196$	2.74713			0.196224
$197$	−14.7803			−1.05305			−0.526527	−	0.850158i	$-0.676506\pi$
$197$	−14.7803			−1.05305			−0.526527	+	0.850158i	$0.676506\pi$
$198$		−	32.4250i		−	2.30434i
$199$		−	12.3298i		−	0.874039i	−0.899452	−	0.437019i	$-0.856034\pi$
$199$		−	12.3298i		−	0.874039i	0.899452	−	0.437019i	$-0.143966\pi$
$200$		0			0
$201$	−21.9707			−1.54970
$202$			8.38279i			0.589811i
$203$			8.84831i			0.621029i
$204$			19.7655i			1.38386i
$205$		0			0
$206$	−13.4727			−0.938689
$207$		−	46.5176i		−	3.23320i
$208$			2.88351i			0.199935i
$209$			0.594082i			0.0410935i
$210$		0			0
$211$	14.1805			0.976224			0.488112	−	0.872781i	$-0.337686\pi$
$211$	14.1805			0.976224			0.488112	+	0.872781i	$0.337686\pi$
$212$	9.87810			0.678431
$213$	11.1750			0.765697
$214$			1.89614i			0.129617i
$215$		0			0
$216$		−	19.0071i		−	1.29327i
$217$			0.877887i			0.0595949i
$218$	17.2044			1.16523
$219$	−3.33718			−0.225506
$220$		0			0
$221$	−16.7452			−1.12641
$222$	16.3619	−	12.6851i	1.09814	−	0.851370i
$223$	7.29071			0.488222			0.244111	−	0.969747i	$-0.421504\pi$
$223$	7.29071			0.488222			0.244111	+	0.969747i	$0.421504\pi$
$224$		−	2.06225i		−	0.137790i
$225$		0			0
$226$	1.17098			0.0778925
$227$			26.3807i			1.75095i	0.483267	+	0.875473i	$0.339450\pi$
$227$			26.3807i			1.75095i	−0.483267	+	0.875473i	$0.660550\pi$
$228$			0.535322i			0.0354526i
$229$	19.5925			1.29471			0.647354	−	0.762190i	$-0.275876\pi$
$229$	19.5925			1.29471			0.647354	+	0.762190i	$0.275876\pi$
$230$		0			0
$231$	−26.5123			−1.74438
$232$	−4.29061			−0.281692
$233$	3.69983			0.242384			0.121192	−	0.992629i	$-0.461328\pi$
$233$	3.69983			0.242384			0.121192	+	0.992629i	$0.461328\pi$
$234$	24.7533			1.61817
$235$		0			0
$236$		−	8.47192i		−	0.551475i
$237$			27.3896i			1.77914i
$238$	11.9760			0.776287
$239$			8.68397i			0.561719i	0.959749	+	0.280860i	$0.0906195\pi$
$239$			8.68397i			0.561719i	−0.959749	+	0.280860i	$0.909380\pi$
$240$		0			0
$241$			27.5610i			1.77536i	0.460464	+	0.887678i	$0.347683\pi$
$241$			27.5610i			1.77536i	−0.460464	+	0.887678i	$0.652317\pi$
$242$		−	3.26714i		−	0.210020i
$243$	−75.5116			−4.84407
$244$		−	1.23904i		−	0.0793215i
$245$		0			0
$246$			3.14401i			0.200455i
$247$	−0.453523			−0.0288570
$248$	−0.425694			−0.0270316
$249$	−40.6716			−2.57746
$250$		0			0
$251$			20.4512i			1.29087i	0.763816	+	0.645434i	$0.223324\pi$
$251$			20.4512i			1.29087i	−0.763816	+	0.645434i	$0.776676\pi$
$252$	−17.7032			−1.11520
$253$		−	20.4679i		−	1.28681i
$254$		−	16.1128i		−	1.01101i
$255$		0			0
$256$	1.00000			0.0625000
$257$			22.8951i			1.42815i	0.700067	+	0.714077i	$0.253153\pi$
$257$			22.8951i			1.42815i	−0.700067	+	0.714077i	$0.746847\pi$
$258$	−36.8483			−2.29407
$259$	−7.68597	−	9.91373i	−0.477583	−	0.616009i
$260$		0			0
$261$			36.8324i			2.27987i
$262$	9.64715			0.596003
$263$	22.1600			1.36644			0.683221	−	0.730212i	$-0.260579\pi$
$263$	22.1600			1.36644			0.683221	+	0.730212i	$0.260579\pi$
$264$		−	12.8560i		−	0.791232i
$265$		0			0
$266$	0.324354			0.0198874
$267$		−	26.0666i		−	1.59525i
$268$	−6.45516			−0.394312
$269$	−23.9657			−1.46121			−0.730607	−	0.682798i	$-0.760763\pi$
$269$	−23.9657			−1.46121			−0.730607	+	0.682798i	$0.760763\pi$
$270$		0			0
$271$	9.82713			0.596956			0.298478	−	0.954417i	$-0.403521\pi$
$271$	9.82713			0.596956			0.298478	+	0.954417i	$0.403521\pi$
$272$			5.80724i			0.352116i
$273$		−	20.2395i		−	1.22495i
$274$			2.53592i			0.153201i
$275$		0			0
$276$		−	18.4435i		−	1.11017i
$277$		−	0.807608i		−	0.0485245i	−0.999706	−	0.0242622i	$-0.992276\pi$
$277$		−	0.807608i		−	0.0485245i	0.999706	−	0.0242622i	$-0.00772366\pi$
$278$			7.56000i			0.453418i
$279$			3.65434i			0.218780i
$280$		0			0
$281$			1.03833i			0.0619414i	0.999520	+	0.0309707i	$0.00985986\pi$
$281$			1.03833i			0.0619414i	−0.999520	+	0.0309707i	$0.990140\pi$
$282$			2.30234i			0.137102i
$283$		−	6.26151i		−	0.372208i	−0.982530	−	0.186104i	$-0.940414\pi$
$283$		−	6.26151i		−	0.372208i	0.982530	−	0.186104i	$-0.0595862\pi$
$284$	3.28329			0.194828
$285$		0			0
$286$	10.8916			0.644031
$287$	1.90497			0.112447
$288$		−	8.58443i		−	0.505842i
$289$	−16.7241			−0.983769
$290$		0			0
$291$		−	47.5041i		−	2.78474i
$292$	−0.980489			−0.0573788
$293$	−4.84496			−0.283045			−0.141523	−	0.989935i	$-0.545200\pi$
$293$	−4.84496			−0.283045			−0.141523	+	0.989935i	$0.545200\pi$
$294$		−	9.35011i		−	0.545309i
$295$		0			0
$296$	4.80724	−	3.72698i	0.279415	−	0.216627i
$297$	−71.7934			−4.16588
$298$			6.72596i			0.389624i
$299$	15.6253			0.903632
$300$		0			0
$301$			22.3265i			1.28688i
$302$			15.8051i			0.909483i
$303$	28.5316			1.63910
$304$			0.157282i			0.00902072i
$305$		0			0
$306$	49.8519			2.84984
$307$	2.87515			0.164093			0.0820467	−	0.996628i	$-0.473854\pi$
$307$	2.87515			0.164093			0.0820467	+	0.996628i	$0.473854\pi$
$308$	−7.78950			−0.443848
$309$			45.8556i			2.60864i
$310$		0			0
$311$			6.07102i			0.344256i	0.985075	+	0.172128i	$0.0550643\pi$
$311$			6.07102i			0.344256i	−0.985075	+	0.172128i	$0.944936\pi$
$312$	9.81429			0.555625
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		−	8.50755i		−	0.480109i
$315$		0			0
$316$			8.04725i			0.452693i
$317$	25.5691			1.43610			0.718052	−	0.695989i	$-0.245034\pi$
$317$	25.5691			1.43610			0.718052	+	0.695989i	$0.245034\pi$
$318$		−	33.6210i		−	1.88537i
$319$			16.2064i			0.907386i
$320$		0			0
$321$	6.45367			0.360209
$322$	−11.1750			−0.622757
$323$	−0.913372			−0.0508214
$324$	−38.9391			−2.16329
$325$		0			0
$326$	5.41321			0.299810
$327$		−	58.5569i		−	3.23820i
$328$			0.923733i			0.0510046i
$329$	1.39500			0.0769087
$330$		0			0
$331$			16.8578i			0.926589i	0.886204	+	0.463295i	$0.153333\pi$
$331$			16.8578i			0.926589i	−0.886204	+	0.463295i	$0.846667\pi$
$332$	−11.9496			−0.655821
$333$	−31.9940	−	41.2674i	−1.75326	−	2.26144i
$334$	2.08165			0.113903
$335$		0			0
$336$	−7.01905			−0.382921
$337$	14.7105			0.801334			0.400667	−	0.916224i	$-0.368779\pi$
$337$	14.7105			0.801334			0.400667	+	0.916224i	$0.368779\pi$
$338$		−	4.68537i		−	0.254851i
$339$		−	3.98554i		−	0.216465i
$340$		0			0
$341$			1.60793i			0.0870741i
$342$	1.35017			0.0730090
$343$	−20.1010			−1.08535
$344$	−10.8263			−0.583715
$345$		0			0
$346$			12.3562i			0.664273i
$347$			3.26296i			0.175165i	0.996157	+	0.0875824i	$0.0279141\pi$
$347$			3.26296i			0.175165i	−0.996157	+	0.0875824i	$0.972086\pi$
$348$			14.6035i			0.782829i
$349$	17.2097			0.921213			0.460607	−	0.887604i	$-0.347632\pi$
$349$	17.2097			0.921213			0.460607	+	0.887604i	$0.347632\pi$
$350$		0			0
$351$		−	54.8072i		−	2.92539i
$352$		−	3.77719i		−	0.201325i
$353$			4.88641i			0.260077i	0.991509	+	0.130039i	$0.0415101\pi$
$353$			4.88641i			0.260077i	−0.991509	+	0.130039i	$0.958490\pi$
$354$	−28.8349			−1.53256
$355$		0			0
$356$		−	7.65857i		−	0.405903i
$357$		−	40.7613i		−	2.15732i
$358$	−13.4571			−0.711229
$359$	−3.39107			−0.178974			−0.0894870	−	0.995988i	$-0.528523\pi$
$359$	−3.39107			−0.178974			−0.0894870	+	0.995988i	$0.528523\pi$
$360$		0			0
$361$	18.9753			0.998698
$362$			3.66608i			0.192685i
$363$	−11.1200			−0.583649
$364$		−	5.94652i		−	0.311682i
$365$		0			0
$366$	−4.21719			−0.220436
$367$	−9.25169			−0.482934			−0.241467	−	0.970409i	$-0.577629\pi$
$367$	−9.25169			−0.482934			−0.241467	+	0.970409i	$0.577629\pi$
$368$		−	5.41883i		−	0.282476i
$369$	7.92972			0.412805
$370$		0			0
$371$	−20.3711			−1.05761
$372$			1.44889i			0.0751214i
$373$	11.4101			0.590792			0.295396	−	0.955375i	$-0.404548\pi$
$373$	11.4101			0.590792			0.295396	+	0.955375i	$0.404548\pi$
$374$	21.9350			1.13423
$375$		0			0
$376$			0.676445i			0.0348850i
$377$	−12.3720			−0.637191
$378$			39.1974i			2.01610i
$379$	−36.3643			−1.86791			−0.933954	−	0.357394i	$-0.883665\pi$
$379$	−36.3643			−1.86791			−0.933954	+	0.357394i	$0.883665\pi$
$380$		0			0
$381$	−54.8415			−2.80961
$382$	−18.5540			−0.949307
$383$			13.6526i			0.697615i	0.937194	+	0.348807i	$0.113413\pi$
$383$			13.6526i			0.697615i	−0.937194	+	0.348807i	$0.886587\pi$
$384$		−	3.40359i		−	0.173689i
$385$		0			0
$386$	−4.89778			−0.249290
$387$			92.9376i			4.72428i
$388$		−	13.9571i		−	0.708562i
$389$		−	2.36833i		−	0.120079i	−0.998196	−	0.0600396i	$-0.980877\pi$
$389$		−	2.36833i		−	0.120079i	0.998196	−	0.0600396i	$-0.0191227\pi$
$390$		0			0
$391$	31.4685			1.59143
$392$		−	2.74713i		−	0.138751i
$393$		−	32.8349i		−	1.65630i
$394$			14.7803i			0.744621i
$395$		0			0
$396$	−32.4250			−1.62942
$397$	34.4978			1.73140			0.865698	−	0.500567i	$-0.166875\pi$
$397$	34.4978			1.73140			0.865698	+	0.500567i	$0.166875\pi$
$398$	−12.3298			−0.618039
$399$		−	1.10397i		−	0.0552675i
$400$		0			0
$401$		−	29.0777i		−	1.45207i	−0.687658	−	0.726035i	$-0.741361\pi$
$401$		−	29.0777i		−	1.45207i	0.687658	−	0.726035i	$-0.258639\pi$
$402$			21.9707i			1.09580i
$403$	−1.22749			−0.0611458
$404$	8.38279			0.417059
$405$		0			0
$406$	8.84831			0.439134
$407$	−14.0775	−	18.1579i	−0.697797	−	0.900051i
$408$	19.7655			0.978537
$409$			16.4980i			0.815773i	0.913033	+	0.407887i	$0.133734\pi$
$409$			16.4980i			0.815773i	−0.913033	+	0.407887i	$0.866266\pi$
$410$		0			0
$411$	8.63124			0.425748
$412$			13.4727i			0.663753i
$413$			17.4712i			0.859702i
$414$	−46.5176			−2.28621
$415$		0			0
$416$	2.88351			0.141376
$417$	25.7311			1.26006
$418$	0.594082			0.0290575
$419$	−22.9409			−1.12074			−0.560368	−	0.828244i	$-0.689340\pi$
$419$	−22.9409			−1.12074			−0.560368	+	0.828244i	$0.689340\pi$
$420$		0			0
$421$			1.72030i			0.0838422i	0.999121	+	0.0419211i	$0.0133478\pi$
$421$			1.72030i			0.0838422i	−0.999121	+	0.0419211i	$0.986652\pi$
$422$		−	14.1805i		−	0.690294i
$423$	5.80690			0.282341
$424$		−	9.87810i		−	0.479723i
$425$		0			0
$426$		−	11.1750i		−	0.541430i
$427$			2.55521i			0.123655i
$428$	1.89614			0.0916532
$429$		−	37.0704i		−	1.78978i
$430$		0			0
$431$		−	2.55447i		−	0.123045i	−0.998106	−	0.0615223i	$-0.980404\pi$
$431$		−	2.55447i		−	0.123045i	0.998106	−	0.0615223i	$-0.0195955\pi$
$432$	−19.0071			−0.914480
$433$	30.0446			1.44385			0.721926	−	0.691970i	$-0.243257\pi$
$433$	30.0446			1.44385			0.721926	+	0.691970i	$0.243257\pi$
$434$	0.877887			0.0421399
$435$		0			0
$436$		−	17.2044i		−	0.823943i
$437$	0.852283			0.0407702
$438$			3.33718i			0.159457i
$439$		−	38.4265i		−	1.83400i	−0.398889	−	0.916999i	$-0.630604\pi$
$439$		−	38.4265i		−	1.83400i	0.398889	−	0.916999i	$-0.369396\pi$
$440$		0			0
$441$	−23.5825			−1.12298
$442$			16.7452i			0.796490i
$443$	−1.00809			−0.0478957			−0.0239479	−	0.999713i	$-0.507624\pi$
$443$	−1.00809			−0.0478957			−0.0239479	+	0.999713i	$0.507624\pi$
$444$	−12.6851	−	16.3619i	−0.602010	−	0.776501i
$445$		0			0
$446$		−	7.29071i		−	0.345225i
$447$	22.8924			1.08277
$448$	−2.06225			−0.0974321
$449$			32.1157i			1.51563i	0.652468	+	0.757817i	$0.273734\pi$
$449$			32.1157i			1.51563i	−0.652468	+	0.757817i	$0.726266\pi$
$450$		0			0
$451$	3.48911			0.164296
$452$		−	1.17098i		−	0.0550783i
$453$	53.7942			2.52747
$454$	26.3807			1.23811
$455$		0			0
$456$	0.535322			0.0250688
$457$			7.85884i			0.367621i	0.982962	+	0.183810i	$0.0588433\pi$
$457$			7.85884i			0.367621i	−0.982962	+	0.183810i	$0.941157\pi$
$458$		−	19.5925i		−	0.915496i
$459$		−	110.379i		−	5.15205i
$460$		0			0
$461$		−	16.7952i		−	0.782233i	−0.920341	−	0.391116i	$-0.872089\pi$
$461$		−	16.7952i		−	0.782233i	0.920341	−	0.391116i	$-0.127911\pi$
$462$			26.5123i			1.23346i
$463$			2.46094i			0.114370i	0.998364	+	0.0571848i	$0.0182124\pi$
$463$			2.46094i			0.114370i	−0.998364	+	0.0571848i	$0.981788\pi$
$464$			4.29061i			0.199187i
$465$		0			0
$466$		−	3.69983i		−	0.171391i
$467$		−	7.57699i		−	0.350621i	−0.984513	−	0.175311i	$-0.943907\pi$
$467$		−	7.57699i		−	0.350621i	0.984513	−	0.175311i	$-0.0560930\pi$
$468$		−	24.7533i		−	1.14422i
$469$	13.3121			0.614698
$470$		0			0
$471$	−28.9562			−1.33423
$472$	−8.47192			−0.389952
$473$			40.8929i			1.88026i
$474$	27.3896			1.25804
$475$		0			0
$476$		−	11.9760i		−	0.548918i
$477$	−84.7979			−3.88263
$478$	8.68397			0.397195
$479$		−	12.6993i		−	0.580246i	−0.956989	−	0.290123i	$-0.906304\pi$
$479$		−	12.6993i		−	0.580246i	0.956989	−	0.290123i	$-0.0936962\pi$
$480$		0			0
$481$	13.8617	−	10.7468i	0.632041	−	0.490012i
$482$	27.5610			1.25537
$483$			38.0351i			1.73066i
$484$	−3.26714			−0.148506
$485$		0			0
$486$			75.5116i			3.42527i
$487$		−	10.8489i		−	0.491611i	−0.969319	−	0.245806i	$-0.920948\pi$
$487$		−	10.8489i		−	0.491611i	0.969319	−	0.245806i	$-0.0790525\pi$
$488$	−1.23904			−0.0560888
$489$		−	18.4243i		−	0.833178i
$490$		0			0
$491$	−17.5266			−0.790965			−0.395482	−	0.918474i	$-0.629423\pi$
$491$	−17.5266			−0.790965			−0.395482	+	0.918474i	$0.629423\pi$
$492$	3.14401			0.141743
$493$	−24.9166			−1.12219
$494$			0.453523i			0.0204050i
$495$		0			0
$496$			0.425694i			0.0191142i
$497$	−6.77096			−0.303719
$498$			40.6716i			1.82254i
$499$			32.4860i			1.45427i	0.686494	+	0.727135i	$0.259149\pi$
$499$			32.4860i			1.45427i	−0.686494	+	0.727135i	$0.740851\pi$
$500$		0			0
$501$		−	7.08508i		−	0.316538i
$502$	20.4512			0.912782
$503$		−	18.1375i		−	0.808712i	−0.914602	−	0.404356i	$-0.867496\pi$
$503$		−	18.1375i		−	0.808712i	0.914602	−	0.404356i	$-0.132504\pi$
$504$			17.7032i			0.788565i
$505$		0			0
$506$	−20.4679			−0.909911
$507$	−15.9471			−0.708235
$508$	−16.1128			−0.714891
$509$	15.2070			0.674037			0.337018	−	0.941498i	$-0.390582\pi$
$509$	15.2070			0.674037			0.337018	+	0.941498i	$0.390582\pi$
$510$		0			0
$511$	2.02201			0.0894485
$512$		−	1.00000i		−	0.0441942i
$513$		−	2.98947i		−	0.131988i
$514$	22.8951			1.00986
$515$		0			0
$516$			36.8483i			1.62215i
$517$	2.55506			0.112371
$518$	−9.91373	+	7.68597i	−0.435584	+	0.337702i
$519$	42.0554			1.84603
$520$		0			0
$521$	21.5850			0.945657			0.472829	−	0.881154i	$-0.343233\pi$
$521$	21.5850			0.945657			0.472829	+	0.881154i	$0.343233\pi$
$522$	36.8324			1.61211
$523$		−	6.64279i		−	0.290469i	−0.989397	−	0.145234i	$-0.953606\pi$
$523$		−	6.64279i		−	0.290469i	0.989397	−	0.145234i	$-0.0463936\pi$
$524$		−	9.64715i		−	0.421438i
$525$		0			0
$526$		−	22.1600i		−	0.966220i
$527$	−2.47211			−0.107687
$528$	−12.8560			−0.559485
$529$	−6.36374			−0.276684
$530$		0			0
$531$			72.7266i			3.15606i
$532$		−	0.324354i		−	0.0140625i
$533$			2.66359i			0.115373i
$534$	−26.0666			−1.12801
$535$		0			0
$536$			6.45516i			0.278820i
$537$			45.8024i			1.97652i
$538$			23.9657i			1.03323i
$539$	−10.3764			−0.446944
$540$		0			0
$541$		−	12.3214i		−	0.529737i	−0.964285	−	0.264868i	$-0.914671\pi$
$541$		−	12.3214i		−	0.529737i	0.964285	−	0.264868i	$-0.0853286\pi$
$542$		−	9.82713i		−	0.422112i
$543$	12.4778			0.535475
$544$	5.80724			0.248983
$545$		0			0
$546$	−20.2395			−0.866171
$547$		−	38.2186i		−	1.63411i	−0.576561	−	0.817054i	$-0.695606\pi$
$547$		−	38.2186i		−	1.63411i	0.576561	−	0.817054i	$-0.304394\pi$
$548$	2.53592			0.108329
$549$			10.6365i			0.453953i
$550$		0			0
$551$	−0.674834			−0.0287489
$552$	−18.4435			−0.785007
$553$		−	16.5954i		−	0.705710i
$554$	−0.807608			−0.0343120
$555$		0			0
$556$	7.56000			0.320615
$557$			32.9212i			1.39492i	0.716626	+	0.697458i	$0.245685\pi$
$557$			32.9212i			1.39492i	−0.716626	+	0.697458i	$0.754315\pi$
$558$	3.65434			0.154701
$559$	−31.2177			−1.32037
$560$		0			0
$561$		−	74.6579i		−	3.15206i
$562$	1.03833			0.0437992
$563$		−	5.39047i		−	0.227181i	−0.993528	−	0.113591i	$-0.963765\pi$
$563$		−	5.39047i		−	0.227181i	0.993528	−	0.113591i	$-0.0362352\pi$
$564$	2.30234			0.0969461
$565$		0			0
$566$	−6.26151			−0.263191
$567$	80.3022			3.37238
$568$		−	3.28329i		−	0.137764i
$569$		−	2.18118i		−	0.0914399i	−0.998954	−	0.0457200i	$-0.985442\pi$
$569$		−	2.18118i		−	0.0914399i	0.998954	−	0.0457200i	$-0.0145582\pi$
$570$		0			0
$571$	41.7356			1.74658			0.873291	−	0.487200i	$-0.161982\pi$
$571$	41.7356			1.74658			0.873291	+	0.487200i	$0.161982\pi$
$572$		−	10.8916i		−	0.455399i
$573$			63.1503i			2.63814i
$574$		−	1.90497i		−	0.0795118i
$575$		0			0
$576$	−8.58443			−0.357685
$577$		−	27.9657i		−	1.16423i	−0.813107	−	0.582114i	$-0.802226\pi$
$577$		−	27.9657i		−	1.16423i	0.813107	−	0.582114i	$-0.197774\pi$
$578$			16.7241i			0.695630i
$579$			16.6700i			0.692783i
$580$		0			0
$581$	24.6431			1.02237
$582$	−47.5041			−1.96911
$583$	−37.3114			−1.54528
$584$			0.980489i			0.0405729i
$585$		0			0
$586$			4.84496i			0.200143i
$587$		−	19.4024i		−	0.800825i	−0.916335	−	0.400412i	$-0.868867\pi$
$587$		−	19.4024i		−	0.800825i	0.916335	−	0.400412i	$-0.131133\pi$
$588$	−9.35011			−0.385592
$589$	−0.0669539			−0.00275879
$590$		0			0
$591$	50.3061			2.06932
$592$	−3.72698	−	4.80724i	−0.153178	−	0.197576i
$593$	−12.5182			−0.514061			−0.257030	−	0.966403i	$-0.582744\pi$
$593$	−12.5182			−0.514061			−0.257030	+	0.966403i	$0.582744\pi$
$594$			71.7934i			2.94572i
$595$		0			0
$596$	6.72596			0.275506
$597$			41.9657i			1.71754i
$598$		−	15.6253i		−	0.638964i
$599$	−11.7177			−0.478771			−0.239386	−	0.970925i	$-0.576946\pi$
$599$	−11.7177			−0.478771			−0.239386	+	0.970925i	$0.576946\pi$
$600$		0			0
$601$	1.37239			0.0559808			0.0279904	−	0.999608i	$-0.491089\pi$
$601$	1.37239			0.0559808			0.0279904	+	0.999608i	$0.491089\pi$
$602$	22.3265			0.909961
$603$	55.4139			2.25663
$604$	15.8051			0.643101
$605$		0			0
$606$		−	28.5316i		−	1.15902i
$607$			24.6378i			1.00002i	0.866021	+	0.500008i	$0.166670\pi$
$607$			24.6378i			1.00002i	−0.866021	+	0.500008i	$0.833330\pi$
$608$	0.157282			0.00637861
$609$		−	30.1160i		−	1.22036i
$610$		0			0
$611$			1.95054i			0.0789102i
$612$		−	49.8519i		−	2.01514i
$613$	27.6733			1.11771			0.558857	−	0.829264i	$-0.311240\pi$
$613$	27.6733			1.11771			0.558857	+	0.829264i	$0.311240\pi$
$614$		−	2.87515i		−	0.116032i
$615$		0			0
$616$			7.78950i			0.313848i
$617$	17.9433			0.722371			0.361185	−	0.932494i	$-0.382372\pi$
$617$	17.9433			0.722371			0.361185	+	0.932494i	$0.382372\pi$
$618$	45.8556			1.84458
$619$	25.1044			1.00903			0.504515	−	0.863403i	$-0.331671\pi$
$619$	25.1044			1.00903			0.504515	+	0.863403i	$0.331671\pi$
$620$		0			0
$621$			102.996i			4.13310i
$622$	6.07102			0.243426
$623$			15.7939i			0.632768i
$624$		−	9.81429i		−	0.392886i
$625$		0			0
$626$	10.0000			0.399680
$627$		−	2.02201i		−	0.0807514i
$628$	−8.50755			−0.339488
$629$	27.9168	−	21.6435i	1.11312	−	0.862983i
$630$		0			0
$631$			47.8987i			1.90682i	0.301679	+	0.953409i	$0.402453\pi$
$631$			47.8987i			1.90682i	−0.301679	+	0.953409i	$0.597547\pi$
$632$	8.04725			0.320102
$633$	−48.2645			−1.91834
$634$		−	25.5691i		−	1.01548i
$635$		0			0
$636$	−33.6210			−1.33316
$637$		−	7.92138i		−	0.313856i
$638$	16.2064			0.641619
$639$	−28.1852			−1.11499
$640$		0			0
$641$	−19.4541			−0.768390			−0.384195	−	0.923252i	$-0.625521\pi$
$641$	−19.4541			−0.768390			−0.384195	+	0.923252i	$0.625521\pi$
$642$		−	6.45367i		−	0.254706i
$643$		−	26.0553i		−	1.02752i	−0.857934	−	0.513760i	$-0.828252\pi$
$643$		−	26.0553i		−	1.02752i	0.857934	−	0.513760i	$-0.171748\pi$
$644$			11.1750i			0.440356i
$645$		0			0
$646$			0.913372i			0.0359362i
$647$			14.7320i			0.579176i	0.957151	+	0.289588i	$0.0935182\pi$
$647$			14.7320i			0.579176i	−0.957151	+	0.289588i	$0.906482\pi$
$648$			38.9391i			1.52967i
$649$			32.0000i			1.25611i
$650$		0			0
$651$		−	2.98797i		−	0.117108i
$652$		−	5.41321i		−	0.211998i
$653$			30.5877i			1.19699i	0.801127	+	0.598494i	$0.204234\pi$
$653$			30.5877i			1.19699i	−0.801127	+	0.598494i	$0.795766\pi$
$654$	−58.5569			−2.28976
$655$		0			0
$656$	0.923733			0.0360657
$657$	8.41694			0.328376
$658$		−	1.39500i		−	0.0543827i
$659$	−29.3533			−1.14344			−0.571720	−	0.820448i	$-0.693724\pi$
$659$	−29.3533			−1.14344			−0.571720	+	0.820448i	$0.693724\pi$
$660$		0			0
$661$		−	16.7963i		−	0.653299i	−0.945145	−	0.326650i	$-0.894080\pi$
$661$		−	16.7963i		−	0.653299i	0.945145	−	0.326650i	$-0.105920\pi$
$662$	16.8578			0.655198
$663$	56.9940			2.21346
$664$			11.9496i			0.463735i
$665$		0			0
$666$	−41.2674	+	31.9940i	−1.59908	+	1.23974i
$667$	23.2501			0.900247
$668$		−	2.08165i		−	0.0805415i
$669$	−24.8146			−0.959388
$670$		0			0
$671$			4.68009i			0.180673i
$672$			7.01905i			0.270766i
$673$	−21.2015			−0.817259			−0.408629	−	0.912700i	$-0.633993\pi$
$673$	−21.2015			−0.817259			−0.408629	+	0.912700i	$0.633993\pi$
$674$		−	14.7105i		−	0.566628i
$675$		0			0
$676$	−4.68537			−0.180207
$677$	−45.9734			−1.76690			−0.883450	−	0.468526i	$-0.844785\pi$
$677$	−45.9734			−1.76690			−0.883450	+	0.468526i	$0.844785\pi$
$678$	−3.98554			−0.153064
$679$			28.7829i			1.10459i
$680$		0			0
$681$		−	89.7890i		−	3.44072i
$682$	1.60793			0.0615707
$683$		−	3.06580i		−	0.117309i	−0.998278	−	0.0586547i	$-0.981319\pi$
$683$		−	3.06580i		−	0.117309i	0.998278	−	0.0586547i	$-0.0186811\pi$
$684$		−	1.35017i		−	0.0516251i
$685$		0			0
$686$			20.1010i			0.767460i
$687$	−66.6848			−2.54418
$688$			10.8263i			0.412749i
$689$		−	28.4836i		−	1.08514i
$690$		0			0
$691$	8.43426			0.320854			0.160427	−	0.987048i	$-0.448713\pi$
$691$	8.43426			0.320854			0.160427	+	0.987048i	$0.448713\pi$
$692$	12.3562			0.469712
$693$	66.8684			2.54012
$694$	3.26296			0.123860
$695$		0			0
$696$	14.6035			0.553543
$697$			5.36434i			0.203189i
$698$		−	17.2097i		−	0.651396i
$699$	−12.5927			−0.476300
$700$		0			0
$701$			20.3110i			0.767136i	0.923513	+	0.383568i	$0.125305\pi$
$701$			20.3110i			0.767136i	−0.923513	+	0.383568i	$0.874695\pi$
$702$	−54.8072			−2.06856
$703$	0.756091	−	0.586186i	0.0285165	−	0.0221084i
$704$	−3.77719			−0.142358
$705$		0			0
$706$	4.88641			0.183902
$707$	−17.2874			−0.650159
$708$			28.8349i			1.08368i
$709$			23.8789i			0.896790i	0.893836	+	0.448395i	$0.148004\pi$
$709$			23.8789i			0.896790i	−0.893836	+	0.448395i	$0.851996\pi$
$710$		0			0
$711$		−	69.0811i		−	2.59074i
$712$	−7.65857			−0.287017
$713$	2.30677			0.0863890
$714$	−40.7613			−1.52545
$715$		0			0
$716$			13.4571i			0.502915i
$717$		−	29.5567i		−	1.10381i
$718$			3.39107i			0.126554i
$719$	−14.6971			−0.548110			−0.274055	−	0.961714i	$-0.588365\pi$
$719$	−14.6971			−0.548110			−0.274055	+	0.961714i	$0.588365\pi$
$720$		0			0
$721$		−	27.7841i		−	1.03473i
$722$		−	18.9753i		−	0.706186i
$723$		−	93.8062i		−	3.48869i
$724$	3.66608			0.136249
$725$		0			0
$726$			11.1200i			0.412702i
$727$			5.61569i			0.208274i	0.994563	+	0.104137i	$0.0332081\pi$
$727$			5.61569i			0.208274i	−0.994563	+	0.104137i	$0.966792\pi$
$728$	−5.94652			−0.220393
$729$	140.193			5.19233
$730$		0			0
$731$	−62.8709			−2.32536
$732$			4.21719i			0.155872i
$733$	16.7565			0.618917			0.309459	−	0.950913i	$-0.399852\pi$
$733$	16.7565			0.618917			0.309459	+	0.950913i	$0.399852\pi$
$734$			9.25169i			0.341486i
$735$		0			0
$736$	−5.41883			−0.199741
$737$	24.3823			0.898135
$738$		−	7.92972i		−	0.291897i
$739$	13.2106			0.485960			0.242980	−	0.970031i	$-0.421875\pi$
$739$	13.2106			0.485960			0.242980	+	0.970031i	$0.421875\pi$
$740$		0			0
$741$	1.54361			0.0567058
$742$			20.3711i			0.747847i
$743$	−23.0799			−0.846721			−0.423360	−	0.905961i	$-0.639150\pi$
$743$	−23.0799			−0.846721			−0.423360	+	0.905961i	$0.639150\pi$
$744$	1.44889			0.0531188
$745$		0			0
$746$		−	11.4101i		−	0.417753i
$747$	102.581			3.75323
$748$		−	21.9350i		−	0.802024i
$749$	−3.91030			−0.142879
$750$		0			0
$751$	25.0437			0.913856			0.456928	−	0.889504i	$-0.348950\pi$
$751$	25.0437			0.913856			0.456928	+	0.889504i	$0.348950\pi$
$752$	0.676445			0.0246674
$753$		−	69.6075i		−	2.53664i
$754$			12.3720i			0.450562i
$755$		0			0
$756$	39.1974			1.42560
$757$		−	33.5882i		−	1.22078i	−0.792099	−	0.610392i	$-0.791012\pi$
$757$		−	33.5882i		−	1.22078i	0.792099	−	0.610392i	$-0.208988\pi$
$758$			36.3643i			1.32081i
$759$			69.6645i			2.52866i
$760$		0			0
$761$	−14.8529			−0.538416			−0.269208	−	0.963082i	$-0.586762\pi$
$761$	−14.8529			−0.538416			−0.269208	+	0.963082i	$0.586762\pi$
$762$			54.8415i			1.98670i
$763$			35.4798i			1.28446i
$764$			18.5540i			0.671261i
$765$		0			0
$766$	13.6526			0.493288
$767$	−24.4289			−0.882075
$768$	−3.40359			−0.122816
$769$		−	8.80892i		−	0.317658i	−0.987306	−	0.158829i	$-0.949228\pi$
$769$		−	8.80892i		−	0.317658i	0.987306	−	0.158829i	$-0.0507718\pi$
$770$		0			0
$771$		−	77.9254i		−	2.80641i
$772$			4.89778i			0.176275i
$773$	−38.8401			−1.39698			−0.698491	−	0.715619i	$-0.746145\pi$
$773$	−38.8401			−1.39698			−0.698491	+	0.715619i	$0.746145\pi$
$774$	92.9376			3.34057
$775$		0			0
$776$	−13.9571			−0.501029
$777$	26.1599	+	33.7423i	0.938481	+	1.21050i
$778$	−2.36833			−0.0849089
$779$			0.145286i			0.00520542i
$780$		0			0
$781$	−12.4016			−0.443764
$782$		−	31.4685i		−	1.12531i
$783$		−	81.5521i		−	2.91443i
$784$	−2.74713			−0.0981118
$785$		0			0
$786$	−32.8349			−1.17118
$787$	−34.2645			−1.22140			−0.610698	−	0.791864i	$-0.709111\pi$
$787$	−34.2645			−1.22140			−0.610698	+	0.791864i	$0.709111\pi$
$788$	14.7803			0.526527
$789$	−75.4234			−2.68514
$790$		0			0
$791$			2.41485i			0.0858623i
$792$			32.4250i			1.15217i
$793$	−3.57279			−0.126873
$794$		−	34.4978i		−	1.22428i
$795$		0			0
$796$			12.3298i			0.437019i
$797$			34.0208i			1.20508i	0.798089	+	0.602539i	$0.205844\pi$
$797$			34.0208i			1.20508i	−0.798089	+	0.602539i	$0.794156\pi$
$798$	−1.10397			−0.0390800
$799$			3.92828i			0.138973i
$800$		0			0
$801$			65.7445i			2.32297i
$802$	−29.0777			−1.02677
$803$	3.70349			0.130693
$804$	21.9707			0.774848
$805$		0			0
$806$			1.22749i			0.0432366i
$807$	81.5694			2.87138
$808$		−	8.38279i		−	0.294905i
$809$		−	23.0866i		−	0.811680i	−0.913944	−	0.405840i	$-0.866979\pi$
$809$		−	23.0866i		−	0.811680i	0.913944	−	0.405840i	$-0.133021\pi$
$810$		0			0
$811$	−14.0477			−0.493280			−0.246640	−	0.969107i	$-0.579327\pi$
$811$	−14.0477			−0.493280			−0.246640	+	0.969107i	$0.579327\pi$
$812$		−	8.84831i		−	0.310515i
$813$	−33.4475			−1.17306
$814$	−18.1579	+	14.0775i	−0.636432	+	0.493417i
$815$		0			0
$816$		−	19.7655i		−	0.691930i
$817$	−1.70278			−0.0595726
$818$	16.4980			0.576839
$819$			51.0474i			1.78374i
$820$		0			0
$821$	−5.87069			−0.204888			−0.102444	−	0.994739i	$-0.532666\pi$
$821$	−5.87069			−0.204888			−0.102444	+	0.994739i	$0.532666\pi$
$822$		−	8.63124i		−	0.301049i
$823$	36.6536			1.27766			0.638832	−	0.769346i	$-0.279418\pi$
$823$	36.6536			1.27766			0.638832	+	0.769346i	$0.279418\pi$
$824$	13.4727			0.469345
$825$		0			0
$826$	17.4712			0.607901
$827$		−	31.2291i		−	1.08594i	−0.839751	−	0.542971i	$-0.817299\pi$
$827$		−	31.2291i		−	1.08594i	0.839751	−	0.542971i	$-0.182701\pi$
$828$			46.5176i			1.61660i
$829$			5.76907i			0.200368i	0.994969	+	0.100184i	$0.0319431\pi$
$829$			5.76907i			0.200368i	−0.994969	+	0.100184i	$0.968057\pi$
$830$		0			0
$831$			2.74877i			0.0953536i
$832$		−	2.88351i		−	0.0999677i
$833$		−	15.9533i		−	0.552747i
$834$		−	25.7311i		−	0.890996i
$835$		0			0
$836$		−	0.594082i		−	0.0205468i
$837$		−	8.09122i		−	0.279673i
$838$			22.9409i			0.792480i
$839$	19.1631			0.661582			0.330791	−	0.943704i	$-0.392684\pi$
$839$	19.1631			0.661582			0.330791	+	0.943704i	$0.392684\pi$
$840$		0			0
$841$	10.5907			0.365195
$842$	1.72030			0.0592854
$843$		−	3.53404i		−	0.121719i
$844$	−14.1805			−0.488112
$845$		0			0
$846$		−	5.80690i		−	0.199645i
$847$	6.73765			0.231508
$848$	−9.87810			−0.339215
$849$			21.3116i			0.731413i
$850$		0			0
$851$	−26.0496	+	20.1959i	−0.892970	+	0.692307i
$852$	−11.1750			−0.382849
$853$		−	30.1823i		−	1.03342i	−0.856160	−	0.516711i	$-0.827156\pi$
$853$		−	30.1823i		−	1.03342i	0.856160	−	0.516711i	$-0.172844\pi$
$854$	2.55521			0.0874376
$855$		0			0
$856$		−	1.89614i		−	0.0648086i
$857$		−	13.4228i		−	0.458513i	−0.973366	−	0.229257i	$-0.926370\pi$
$857$		−	13.4228i		−	0.458513i	0.973366	−	0.229257i	$-0.0736295\pi$
$858$	−37.0704			−1.26556
$859$			25.1243i			0.857230i	0.903487	+	0.428615i	$0.140998\pi$
$859$			25.1243i			0.857230i	−0.903487	+	0.428615i	$0.859002\pi$
$860$		0			0
$861$	−6.48373			−0.220965
$862$	−2.55447			−0.0870057
$863$	−6.80983			−0.231809			−0.115905	−	0.993260i	$-0.536977\pi$
$863$	−6.80983			−0.231809			−0.115905	+	0.993260i	$0.536977\pi$
$864$			19.0071i			0.646635i
$865$		0			0
$866$		−	30.0446i		−	1.02096i
$867$	56.9219			1.93317
$868$		−	0.877887i		−	0.0297974i
$869$		−	30.3960i		−	1.03111i
$870$		0			0
$871$			18.6135i			0.630695i
$872$	−17.2044			−0.582616
$873$			119.813i			4.05507i
$874$		−	0.852283i		−	0.0288289i
$875$		0			0
$876$	3.33718			0.112753
$877$	−29.4096			−0.993090			−0.496545	−	0.868011i	$-0.665398\pi$
$877$	−29.4096			−0.993090			−0.496545	+	0.868011i	$0.665398\pi$
$878$	−38.4265			−1.29683
$879$	16.4903			0.556202
$880$		0			0
$881$	50.5602			1.70342			0.851708	−	0.524016i	$-0.175567\pi$
$881$	50.5602			1.70342			0.851708	+	0.524016i	$0.175567\pi$
$882$			23.5825i			0.794066i
$883$		−	40.4904i		−	1.36261i	−0.732000	−	0.681305i	$-0.761413\pi$
$883$		−	40.4904i		−	1.36261i	0.732000	−	0.681305i	$-0.238587\pi$
$884$	16.7452			0.563203
$885$		0			0
$886$			1.00809i			0.0338674i
$887$	8.64379			0.290230			0.145115	−	0.989415i	$-0.453645\pi$
$887$	8.64379			0.290230			0.145115	+	0.989415i	$0.453645\pi$
$888$	−16.3619	+	12.6851i	−0.549069	+	0.425685i
$889$	33.2287			1.11445
$890$		0			0
$891$	147.080			4.92738
$892$	−7.29071			−0.244111
$893$			0.106392i			0.00356028i
$894$		−	22.8924i		−	0.765637i
$895$		0			0
$896$			2.06225i			0.0688949i
$897$	−53.1820			−1.77569
$898$	32.1157			1.07171
$899$	−1.82649			−0.0609168
$900$		0			0
$901$		−	57.3645i		−	1.91109i
$902$		−	3.48911i		−	0.116175i
$903$		−	75.9903i		−	2.52880i
$904$	−1.17098			−0.0389462
$905$		0			0
$906$		−	53.7942i		−	1.78719i
$907$		−	33.0245i		−	1.09656i	−0.836294	−	0.548281i	$-0.815283\pi$
$907$		−	33.0245i		−	1.09656i	0.836294	−	0.548281i	$-0.184717\pi$
$908$		−	26.3807i		−	0.875473i
$909$	−71.9615			−2.38681
$910$		0			0
$911$		−	0.940972i		−	0.0311758i	−0.999879	−	0.0155879i	$-0.995038\pi$
$911$		−	0.940972i		−	0.0311758i	0.999879	−	0.0155879i	$-0.00496198\pi$
$912$		−	0.535322i		−	0.0177263i
$913$	45.1360			1.49378
$914$	7.85884			0.259947
$915$		0			0
$916$	−19.5925			−0.647354
$917$			19.8948i			0.656985i
$918$	−110.379			−3.64305
$919$		−	54.5548i		−	1.79960i	−0.436306	−	0.899798i	$-0.643713\pi$
$919$		−	54.5548i		−	1.79960i	0.436306	−	0.899798i	$-0.356287\pi$
$920$		0			0
$921$	−9.78583			−0.322454
$922$	−16.7952			−0.553122
$923$		−	9.46740i		−	0.311623i
$924$	26.5123			0.872189
$925$		0			0
$926$	2.46094			0.0808716
$927$		−	115.656i		−	3.79863i
$928$	4.29061			0.140846
$929$	22.7416			0.746128			0.373064	−	0.927806i	$-0.378307\pi$
$929$	22.7416			0.746128			0.373064	+	0.927806i	$0.378307\pi$
$930$		0			0
$931$		−	0.432073i		−	0.0141606i
$932$	−3.69983			−0.121192
$933$		−	20.6633i		−	0.676485i
$934$	−7.57699			−0.247927
$935$		0			0
$936$	−24.7533			−0.809087
$937$	−43.2122			−1.41168			−0.705841	−	0.708370i	$-0.749431\pi$
$937$	−43.2122			−1.41168			−0.705841	+	0.708370i	$0.749431\pi$
$938$		−	13.3121i		−	0.434657i
$939$		−	34.0359i		−	1.11072i
$940$		0			0
$941$	32.0847			1.04593			0.522966	−	0.852354i	$-0.324825\pi$
$941$	32.0847			1.04593			0.522966	+	0.852354i	$0.324825\pi$
$942$			28.9562i			0.943444i
$943$		−	5.00555i		−	0.163003i
$944$			8.47192i			0.275737i
$945$		0			0
$946$	40.8929			1.32954
$947$		−	8.26212i		−	0.268483i	−0.990949	−	0.134241i	$-0.957140\pi$
$947$		−	8.26212i		−	0.268483i	0.990949	−	0.134241i	$-0.0428598\pi$
$948$		−	27.3896i		−	0.889571i
$949$			2.82725i			0.0917764i
$950$		0			0
$951$	−87.0268			−2.82204
$952$	−11.9760			−0.388144
$953$	−51.1883			−1.65815			−0.829076	−	0.559136i	$-0.811133\pi$
$953$	−51.1883			−1.65815			−0.829076	+	0.559136i	$0.811133\pi$
$954$			84.7979i			2.74543i
$955$		0			0
$956$		−	8.68397i		−	0.280860i
$957$		−	55.1601i		−	1.78307i
$958$	−12.6993			−0.410296
$959$	−5.22970			−0.168876
$960$		0			0
$961$	30.8188			0.994154
$962$	−10.7468	−	13.8617i	−0.346491	−	0.446920i
$963$	−16.2772			−0.524527
$964$		−	27.5610i		−	0.887678i
$965$		0			0
$966$	38.0351			1.22376
$967$			1.84215i			0.0592395i	0.999561	+	0.0296197i	$0.00942963\pi$
$967$			1.84215i			0.0592395i	−0.999561	+	0.0296197i	$0.990570\pi$
$968$			3.26714i			0.105010i
$969$	3.10875			0.0998673
$970$		0			0
$971$	52.6384			1.68925			0.844624	−	0.535361i	$-0.179824\pi$
$971$	52.6384			1.68925			0.844624	+	0.535361i	$0.179824\pi$
$972$	75.5116			2.42203
$973$	−15.5906			−0.499811
$974$	−10.8489			−0.347621
$975$		0			0
$976$			1.23904i			0.0396608i
$977$		−	47.7629i		−	1.52807i	−0.645174	−	0.764035i	$-0.723215\pi$
$977$		−	47.7629i		−	1.52807i	0.645174	−	0.764035i	$-0.276785\pi$
$978$	−18.4243			−0.589146
$979$			28.9278i			0.924538i
$980$		0			0
$981$			147.690i			4.71539i
$982$			17.5266i			0.559297i
$983$	−34.6640			−1.10561			−0.552805	−	0.833310i	$-0.686443\pi$
$983$	−34.6640			−1.10561			−0.552805	+	0.833310i	$0.686443\pi$
$984$		−	3.14401i		−	0.100227i
$985$		0			0
$986$			24.9166i			0.793507i
$987$	−4.74800			−0.151131
$988$	0.453523			0.0144285
$989$	58.6659			1.86547
$990$		0			0
$991$			45.9061i			1.45826i	0.684378	+	0.729128i	$0.260074\pi$
$991$			45.9061i			1.45826i	−0.684378	+	0.729128i	$0.739926\pi$
$992$	0.425694			0.0135158
$993$		−	57.3771i		−	1.82081i
$994$			6.77096i			0.214762i
$995$		0			0
$996$	40.6716			1.28873
$997$		−	24.7232i		−	0.782993i	−0.920180	−	0.391496i	$-0.871958\pi$
$997$		−	24.7232i		−	0.782993i	0.920180	−	0.391496i	$-0.128042\pi$
$998$	32.4860			1.02832
$999$	70.8392	+	91.3718i	2.24125	+	2.89088i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.d.i.1701.1		20
5.2	odd	4		370.2.c.b.369.10	yes	10
5.3	odd	4		370.2.c.a.369.1	✓	10
5.4	even	2	inner	1850.2.d.i.1701.20		20
15.2	even	4		3330.2.e.c.739.6		10
15.8	even	4		3330.2.e.d.739.6		10
37.36	even	2	inner	1850.2.d.i.1701.11		20
185.73	odd	4		370.2.c.b.369.1	yes	10
185.147	odd	4		370.2.c.a.369.10	yes	10
185.184	even	2	inner	1850.2.d.i.1701.10		20
555.332	even	4		3330.2.e.d.739.5		10
555.443	even	4		3330.2.e.c.739.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.c.a.369.1	✓	10	5.3	odd	4
370.2.c.a.369.10	yes	10	185.147	odd	4
370.2.c.b.369.1	yes	10	185.73	odd	4
370.2.c.b.369.10	yes	10	5.2	odd	4
1850.2.d.i.1701.1		20	1.1	even	1	trivial
1850.2.d.i.1701.10		20	185.184	even	2	inner
1850.2.d.i.1701.11		20	37.36	even	2	inner
1850.2.d.i.1701.20		20	5.4	even	2	inner
3330.2.e.c.739.5		10	555.443	even	4
3330.2.e.c.739.6		10	15.2	even	4
3330.2.e.d.739.5		10	555.332	even	4
3330.2.e.d.739.6		10	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.40359 q^{3} -1.00000 q^{4} +3.40359i q^{6} +2.06225 q^{7} +1.00000i q^{8} +8.58443 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -3.40359 q^{3} -1.00000 q^{4} +3.40359i q^{6} +2.06225 q^{7} +1.00000i q^{8} +8.58443 q^{9} +3.77719 q^{11} +3.40359 q^{12} +2.88351i q^{13} -2.06225i q^{14} +1.00000 q^{16} +5.80724i q^{17} -8.58443i q^{18} +0.157282i q^{19} -7.01905 q^{21} -3.77719i q^{22} -5.41883i q^{23} -3.40359i q^{24} +2.88351 q^{26} -19.0071 q^{27} -2.06225 q^{28} +4.29061i q^{29} +0.425694i q^{31} -1.00000i q^{32} -12.8560 q^{33} +5.80724 q^{34} -8.58443 q^{36} +(-3.72698 - 4.80724i) q^{37} +0.157282 q^{38} -9.81429i q^{39} +0.923733 q^{41} +7.01905i q^{42} +10.8263i q^{43} -3.77719 q^{44} -5.41883 q^{46} +0.676445 q^{47} -3.40359 q^{48} -2.74713 q^{49} -19.7655i q^{51} -2.88351i q^{52} -9.87810 q^{53} +19.0071i q^{54} +2.06225i q^{56} -0.535322i q^{57} +4.29061 q^{58} +8.47192i q^{59} +1.23904i q^{61} +0.425694 q^{62} +17.7032 q^{63} -1.00000 q^{64} +12.8560i q^{66} +6.45516 q^{67} -5.80724i q^{68} +18.4435i q^{69} -3.28329 q^{71} +8.58443i q^{72} +0.980489 q^{73} +(-4.80724 + 3.72698i) q^{74} -0.157282i q^{76} +7.78950 q^{77} -9.81429 q^{78} -8.04725i q^{79} +38.9391 q^{81} -0.923733i q^{82} +11.9496 q^{83} +7.01905 q^{84} +10.8263 q^{86} -14.6035i q^{87} +3.77719i q^{88} +7.65857i q^{89} +5.94652i q^{91} +5.41883i q^{92} -1.44889i q^{93} -0.676445i q^{94} +3.40359i q^{96} +13.9571i q^{97} +2.74713i q^{98} +32.4250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{4} + 16 q^{9}+O(q^{10}) \) 20 * q - 20 * q^4 + 16 * q^9	\( 20 q - 20 q^{4} + 16 q^{9} + 20 q^{16} - 24 q^{21} + 4 q^{26} + 36 q^{34} - 16 q^{36} - 8 q^{41} - 20 q^{46} + 16 q^{49} - 20 q^{64} - 40 q^{71} - 16 q^{74} + 116 q^{81} + 24 q^{84} + 20 q^{86} + 164 q^{99}+O(q^{100}) \) 20 * q - 20 * q^4 + 16 * q^9 + 20 * q^16 - 24 * q^21 + 4 * q^26 + 36 * q^34 - 16 * q^36 - 8 * q^41 - 20 * q^46 + 16 * q^49 - 20 * q^64 - 40 * q^71 - 16 * q^74 + 116 * q^81 + 24 * q^84 + 20 * q^86 + 164 * q^99

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